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SELF-TAUGHT 

MECHANICAL DRAWING 

AND ELEMENTARY 

MACHINE DESIGN 



A Treatise 

Comprising- the First Principles of Geometric and Mechanical 
Drawing, Workshop Mathematics, Mechanics, Strength 
of Materials, and the Design of Machine Details, 
including Cams, Sprockets, Gearing, Shafts, 
Pulleys, Belting, Couplings, Screws and 
Bolts, Clutches, Flywheels, etc. Pre¬ 
pared for the Use of Practical 
Mechanics and Young 
Draftsmen. 

By F. L. SYLVESTER, M.E. / 

With Additions 

By ERIK OBERG 

Associate Editor of “Machinery” 

With Chapters on the “Technique of Mechanical Drawing” 
and “ Freehand Lettering for Working Drawing^ ’’ 

By C. W. REINHARDT 



FULLY ILLUSTRATED 


NEW YORK 

THE NORMAN W. HENLEY PUBLISHING CO. 
2 WEST 45th STREET 
1923 
















iJ ^30 

,S q 

/9g,3 


Copyrighted 1923 and 1910, by 
The Norman W. Henley Publishing Co. 


Printed in U.S.A. 




OCT 23 

^ y 


©C1A759521 

"VUe j 








PREFACE 


The demand for an elementary treatise on 
mechanical drawing, including the first principles 
of machine design, and presented in such a way 
as to meet, in particular, the needs of the student 
whose previous theoretical knowledge is limited, 
has caused the author to prepare the present vol¬ 
ume. It has been the author’s aim to adapt this 
treatise to the requirements of the practical me¬ 
chanic and young draftsman, and to present the 
matter in as clear and concise a manner as possible, 
so as to make ‘‘self-study” easy. In order to meet 
the demands of this class of students, practically 
all the important elements of machine design have 
been dealt with, and, besides, algebraic formulas 
have been explained and the elements of trigo¬ 
nometry have been treated in a manner suited to 
the needs of the practical man. 

In arranging the material, the author has first 
devoted himself to mechanical drawing, pure and 
simple, because a thorough understanding of the 
principles of representing objects greatly facilitates 
further study of mechanical subjects; then, atten¬ 
tion has been given to the mathematics necessary 


111 


IV 


PREFACE 


for the solution of the problems in machine design 
presented later, and to a practical introduction to 
theoretical mechanics and strength of materials; 
and, finally, the various elements entering in ma¬ 
chine design, such as cams, gears, sprocket wheels, 
cone pulleys, bolts, screws, couplings, clutches, 
shafting, fly-wheels, etc., have been treated. This 
arrangement makes it possible to present a con- . 
tinuous course of study which is easily compre¬ 
hended and assimilated even by students of limited 
previous training. 

Portions of the section on mechanical drawing 
were published by the author in The Patternmaker 
several years ago. These articles have, however, 
been carefully revised to harmonize with the pres¬ 
ent treatise, and in some sections amplified. In 
the preparation of the material, the author has 
also consulted the works of various authors on 
machine design, and credit has been given in the 
text wherever use has been made of material from 
such sources. 

Several important additions have been made by 
Mr. Erik Oberg, Associate Editor of Machinery. 

In the preparation of these additions, use has partly 
been made of material published from time to time 
in Machinery. 

In the present edition special chapters on the 
Technique of Mechanical Drawing and Freehand 
Lettering for Working Drawings by C. W. Rein¬ 
hardt have also been added. 


OCTOBEK, 1023. 


The Publisher. 


CONTENTS 


Preface 


Page iii 


CHAPTER I 

INSTRUMENTS AND MATERIALS 

General Remarks on the Study of Drawing—Drawing 
Instruments—Pencils—Use of the Instruments— 
Paper—Ink. Page 1 


CHAPTER II 

DEFINITIONS OF TERMS USED IN GEOMETRICAL AND 

MECHANICAL DRAWING 

Point — Line — Surface—Solid—Plane—Angle—Circle 
—Parallelogram — Polygon—Ellipse — Involute — 
Cycloid—Parabola. Page 10 


CHAPTER III 

GEOMETRICAL PROBLEMS 

Bisecting of Lines and Angles—Perpendicular Lines— 
Tangents—Regular Polygons—Inscribed and Cir¬ 
cumscribed Ci rcles—Ellipses—Spirals—Involutes 
—Cycloids—Parabolas. Page 17 


V 






VI 


CONTENTS 


CHAPTER IV 

PROJECTION 

Mode of Representing Objects—Projections of Inclined 
Prisms—Surface Developments of Cones and Pyra¬ 
mids—Intersecting Cylinders, and Cylinder and 
Cone—Projection of a Helix—Isometric Projec¬ 
tion. Page 32 


CHAPTER V 

WORKING DRAWINGS 

Object of Working Drawings—Assembly Drawings— 
Detail Drawings—Dimensions—Finish Marks— 
Sectional Views—Cross-section Chart—Screw 
Threads—Shade Lines—Tracing and Blue-print¬ 
ing. Page 50 


CHAPTER VI 

ALGEBRAIC FORMULAS 

The Meaning of Formulas—Square and Square Root— 
Cube and Cube Roots — Exponents—Areas and 
Volumes of Plane Figures and Solids. Page 79 


CHAPTER VH 

ELEMENTS OF TRIGONOMETRY 

Angles — Right-angled Triangles — Trigonometrical 
Functions—Tables of Natural Functions—Solution 
of Right-angled Triangles—Solution of Oblique- 
angled Triangles—Laying Out Angles by Means 
of Trigonometric Functions. Page 96 






CONTENTS 


vii 


CHAPTER VIII 

ELEMENTS OF MECHANICS 

Resolution of Forces—Levers — Fixed and Movable 
Pulleys—Inclined Planes—The Screw—Differential 
Screw—Newton's Laws of Motion—Pendulum— 
Falling Bodies—Energy and Work—Horse-power 
of Steam Engines. Page 120 


CHAPTER IX 

FIRST PRINCIPLES OF STRENGTH OF MATERIALS 

Factor of Safety—Shape of Machine Parts—Strength 
of Materials as Given by Kirkaldy’s Tests—Stresses 
in Castings. Page 151 


CHAPTER X 
CAMS 

General Principles—Design of Cams Imparting Uniform 
Motion — Reciprocating Cams — Cams Providing 
Uniform Return—Uniformly Accelerated Motion 
Cams — Gravity Cam Curve — Harmonic Action 
Cams—Approximate Gravity Cam Curve. .Page 164 


CHAPTER XI 

SPROCKET WHEELS 

Object of Sprocket Wheels — Drafting of Sprocket 
Wheels for Different Classes of Chain—Speed 
Ratio.. Page 185 





Vlll 


CONTENTS 


CHAPTER XII 

GENERAL PRINCIPLES OF GEARING 

Friction and Knuckle Gearing—Epicycloidal Gearing 
—Gears with Strengthened Flanks—Gears with 
Radial Flanks^—Involute Gears—Interference in 
Involute Gears—The Two Systems Compared— 
Twenty-degree Involute Gears—Shrouded Gears— 
Bevel Gears—Worm Gearing—Circular Pitch— 
Proportions of Teeth — Diametral Pitch — The 
Hunting Tooth—Approximate Shapes for Cycloidal 
Gear Teeth—Involute Teeth—Proportions of Gears 
—Strength of Gear Teeth—Thurston’s Rule for 
Gear Shafts—Speed Ratio of Gearing. . .Page 190 

CHAPTER XIII 

CALCULATING THE DIMENSIONS OF GEARS 

Spur Gearing—Bevel Gears—Worm Gearing. . Page 222 

CHAPTER XIV 

CONE PULLEYS 

Conical Drums — Influence of Crossed Belt — Cone 
Pulleys — Smith’s Rule for Laying Out Cone 
Pulleys. Page 239 


CHAPTER Xy 

BOLTS, STUDS AND SCREWS 

Kinds of Screws—United States Standard Screw Thread * 
—Check or Lock Nuts—Bolts to Withstand Shock 
—Wrench Action—Screws for Power Transmission 
—Efficiency of Screws—Acme Standard Thread— 
Miscellaneous Screw Thread Systems—Other Com¬ 
mercial Forms of Screws. Page 




CONTENTS 


IX 


CHAPTER XVI 

COUPLINGS AND CLUTCHES 

Simple Eorms of Couplings—Calculation of Flange 
Coupling Bolts — Oldham’s Coupling — Hooke’s 
Coupling or Universal Joint—Toothed Clutches— 
Friction Clutches—Cone Clutches. Page 259 


CHAPTER XVII 

SHAFTS, BELTS AND PULLEYS 

Calculation of Shafting—Horse-power of Belting— 
Speed of Belting—Pulley Sizes and Speed Ratios 
—Twisted and Unusual Cases of Belting. . Page 272 


CHAPTER XVIII 

FLY-WHEELS FOR PRESSES, PUNCHES, ETC. 

Object of Fly-wheels—Formulas for Fly-wheel Calcu¬ 
lations— Example of Fly-wheel Calculation for 
Shears. Page 289 


CHAPTER XIX 

TRAINS OF MECHANISM 

To Secure Increase, of Speed—To Secure Reversal of 
Direction—The Compound Idler—The Screw Cut¬ 
ting Train—Simplified Rules for Calculating Lathe 
Change Gears—Back-Gears. Page 297 





X 


CONTENTS 


CHAPTER XX 

QUICK RETURN MOTIONS 

Object of Quick Return Motions—Examples of Simple 
Designs of Quick Return Motions—The Whitworth 
Quick Return Device—The Elliptic Gear Quick 
Return. Page 313 


CHAPTER XXI 

THE TECHNIQUE OF MECHANICAL DRAWING 

Advantage of Facsimile Reproductions and Well 
Executed Drawings—Outline Shading—Section 
Lining—Curved Surface Shading—Shapes of Breaks 
of Different Materials and Objects. Page 326 

CHAPTER XXII 

FREEHAND LETTERING FOR WORKING DRAWINGS 

One-Stroke Lettering—Methods of Construction— 
Proportions—Standard Sizes—Progressive Arrange¬ 
ment of Groups—Slanting Lettering—Vertical 
Lettering—Dimensions—Construction of Titles— 
Spacing of Letters. Page 332 





SELF-TAUGHT 
MECHANICAL DRAWING 


CHAPTER I 

INSTRUMENTS AND MATERIALS 

One who is to study the subject of drawing 
should not merely read a book on the subject, but 
should prepare sheets of exercises. This will fix 
the principles which he learns in his mind in a way 
as reading alone will not do, and will give him 
practical experience in the use of the tools. The 
geometrical problems given in this book make 
perhaps the best of subjects for a beginning, as 
their proper execution will require careful work. 
Later, the student may make dimensioned free¬ 
hand sketches of some machine with which he is 
familiar, and from these sketches he may make up 
a set of finished working drawings. In all of this 
work, care should be taken to have it so laid out, 
with proper margins and spaces between different 
parts, that the drawing when finished shall pre¬ 
sent an appearance of neatness and methodical 
arrangement. 

For the purposes of the student, a drawing board 
about 15 by 18 inches will be large enough. With 
this should be an 18-inch T-square, a pair of 6-inch 
triangles, and a set of three or four irregular curves. 

1 


2 


SELF-TAUGHT MECHANICAL DRAWING 


For drawing full-size work, a good flat beveled- 
edge rule will answer ordinary requirements, but 
for making half- or quarter-size drawings some 
kind of a scale’^ will be found desirable. The tri¬ 
angular scale shown in Fig. 1 is perhaps the one 
mostly used, and it has the advantage of possess- 



Fig. 1. —The Triangular Scale. 


ing six surfaces for graduations, giving variety 
enough for all sorts of conditions, but it has the 
disadvantage of persistently presenting the wrong 
edge, and putting one to the trouble of turning it 
over and over to get the desired edge. This trouble 
may, of course, be overcome by using a scale guard 
such as is shown in Fig. 2, but the guard is itself 

often in the way. As 
but two or three differ¬ 
ent scales, aside from 
full size, will be likely 
to be required, it will be 
found much more con- 
Fig. 2.-Scale Guard or Holder venient to have a sep- 
used on Triangular Scale. arate flat scale for each 

graduation. Such scales 
may be purchased, or, if one is satisfied with the 
open graduation system shown in Fig. 3, he may 
make them without much trouble himself. In this 
system, only one inch is divided, this inch being 
numbered 0; and measurements which include a 



















INSTRUMENTS AND MATERIALS 


3 


fractional part of an inch are reckoned from the 
required whole number to the proper place on the 
divided inch. 

The drawing instruments themselves, while not 
necessarily of the highest price, should be of a 
good serviceable quality of German silver. The 
cheap brass or nickel plated school sets should not 
be considered, as they will prove unsatisfactory 
for regular work. It is not necessary to have a 
large number of instruments. A very good set, 
sufficient for all ordinary requirements, might be 
as follows: First a pair of about 4i- or 5-inch com- 


















- \ - 

1 




012345t) 78' 

10 n 12 13 14 la IS 17 18 


« 

Fig. 3.—Inexpensive Type of Scale. 


passes with fixed needle points (bayonet points are 
useless) and interchangeable pin and pencil points, 
with lengthening bar. Then, a pair of hair¬ 
spring spacers of about the same size. These re¬ 
semble ordinary plain compasses, but the steel end 
of one leg is made adjustable by means of a 
thumb screw. Next, a pair of ruling pens, one 
large and one small, and, lastly, a set of three 
spring instruments, pen, pencil and spacers, for 
small work. Rather than to get cheap instru¬ 
ments, it would be advisable to obtain a set gradu¬ 
ally by getting the large instruments and one pen 
first, and adding the second pen and the spring 
instruments later. The large compasses can, if 
necessary, be used to make circles of from about 
i inch to about 18 or 20 inches in diameter, so 





















4 


SELF-TAUGHT MECHANICAL DRAWING 


that they will do very well for a beginning. For 
making larger circles, beam compasses, in which 
separate heads for the needle point and for the pen 
or pencil point are attached to a wooden bar, after 
the manner of workmen’s trammels, are used. 

A convenient case for the instruments, when 
they are bought separately, is shown in Fig. 4, 
and is made as follows: Take two pieces of 
chamois skin or thin broadcloth, one of them about 
one-half longer than the longest instrument, and 
somewhat wider than all of them when they are 



laid out side by side, and the second one of the 
same width as the first, but somewhat shorter than 
the longest instrument. This second piece is 
sewed onto the large piece at one end by the outer 
edges. Pockets for the reception of the instru¬ 
ments are then made as shown, and when the free 
end of the large piece is folded over, the instru¬ 
ments are rolled up together. 

The pencils, which to avoid scratching particles, 
should be of best quality, should not be sharpened 
to a round point, but to a flat oval point, as such 
a shape will wear longer than a round point; the 
leads used in the compasses, however, should be 










INSTRUMENTS AND MATERIALS 


5 


only slightly flattened. It will be found desirable 
to have two grades of pencils, one quite hard, 
about “4H, to be used for laying out work, and 
a softer one, about “2H,’’ to be used for going 
over the lines of work which is not to be inked in. 
In laying-out work where the hard pencil is used, 
only a moderate pressure should be applied, so as 
to permit of erasures at any time, whether for the 
purpose of making alterations, or to free the draw¬ 
ing of pencil marks after inking. 

The drawing pens should be kept sharp, though 
not so sharp as to cut the paper, and their ends 
should present a neat oval shape. The needle 
points of the compasses should also be kept sharp 
to avoid the tendency to slip when doing work 
where it is undesirable to prick through the pa¬ 
per. A small Arkansas stone will be found useful 
for this purpose. Where much use is made of a 
given center, it may be desirable to employ a horn 
or metal center, such as are kept in stock by deal¬ 
ers in artists^ supplies, to avoid the troublesome 
enlargement of the center in the paper which the 
points of the compasses would otherwise make. 

In making a drawing, care should be taken to 
have the preliminary pencil work done correctly. 
It is a mistake which beginners are likely to make, 
to think that errors in the pencil work may be 
readily corrected in the inking. This, however, 
is usually another case where ‘'haste makes 
waste. It is much better to spend a little extra 
time on the pencil work, than to have to throw 
away a nearly finished ink drawing and do the 
work all over again. In locating the various 


n SELF-TAUGHT MECHANICAL DRAWING 

views of a drawing upon the paper, it will fre¬ 
quently be found to be well to make rough sketches 
of it on scrap paper. These sketches can then be 
moved around on the drawing paper until the best 
arrangement is secured. 

In making a drawing, it will be found most con¬ 
venient, ordinarily, to limit the use of the T-square 
to horizontal lines, the head of the square being 
kept pressed firmly against the left-hand end of 
the drawing board. Vertical lines are then made 



with the aid of the triangles resting against the 
blade of the T-square. Vertical lines which are 
too long to be made in this way, are, of course, 
made with the T-square itself. In inking in a 
drawing, it is best to draw all curved or circular 
lines first, as it is easier to join straight lines onto 
curved lines than to join curved lines onto straight 
lines. Care should also be taken to have meeting 
lines just meet, whether they meet end to end or 
at an angle. Carelessness in this respect gives a 
drawing a very bad appearance, as shown by Fig. 
5, A and B. 






INSTRUMENTS AND MATERIALS 7 

In using the pens, whether the ruling or the com¬ 
pass pens, care should be taken to see that both nibs 
rest upon the paper, otherwise lines such as shown 
in Fig. 6 may result. If the pen does rest squarely 
upon the paper, and such lines continue to appear, 
it is fair to infer that the paper has become some¬ 
what greasy, perhaps from too much handling. 
This trouble may be avoided, and the work kept 
cleaner, by having a piece of thin paper inter¬ 
posed between the hands and the drawing paper. 

The cross hatching work, such as is shown at A 
in Fig. 5, is frequently done by simply using one 
of the triangles resting against the blade of the 


Fig. 6.—Line Resulting from not Having both Pen Points or 
Nibs Resting on the Paper when Inking. 

T-square, the same as is done for vertical lines, the 
spacing being done entirely by the eye; but unless 
one is doing a good deal of this work, so as to 
keep in practice, he will find it very difficult to 
make the spacing regular. There are various sec¬ 
tion-lining devices on the market for doing this 
work, some of them quite expensive. Fig. 7 shows 
a sirnple device for cross-sectioning, which serves 
the purpose as well as any of the more elaborate 
ones, and possesses the additional advantage that 
anyone may readily make it for himself. This 
instrument was shown by Mr. E. W. Beardsley in 
Machinery, September, 1905. An old instrument 
screw, B, is screwed into a slightly smaller hole 
in a piece of wood. A, shaped as shown, and of a 





8 


SELF-TAUGHT MECHANICAL DRAWING 


thickness a little in excess of the diameter of the 
screw-head. This combination is then used in the 
central hole in a triangle, as shown. Then, with 
one finger on the triangle itself, and with another 
one on A, the two may be moved along, first one 
and then the other, for section lining, the desired 
width of space being secured by the adjustment 
given to B. 

For making erasures of ink lines on paper, a 
steel scraping eraser ora sharp knife blade is usu- 



Fig. 7.—Simple Cross-section Liner. 


ally the best, the roughened surface being after¬ 
wards rubbed down smooth with some hard sub¬ 
stance. When making erasures of either pencil 
or ink with a rubber eraser, an erasing shield, 
such as is shown in Fig. 8, is useful for prevent¬ 
ing rubbing out more than is intended. These 
shields are made both of thin sheet metal and of 
celluloid; the metal ones, being the thinner, are 
the more convenient to use. 

The paper used, if good work is desired, should 










INSTRUMENTS AND MATERIALS 9 

be regular drawing paper, whether it be white or 
brown. This has an unglazed surface, and will be 
found much more satisfactory in every way than 
common paper. The glazed surface of the cheaper 
paper does not take pencil marks well, and is torn 
up badly in making erasures. Such paper, if used 
at all, should be used only on the most temporary 



Fig. 8.—Erasing Shield made from Sheet Metal 

or Celluloid. 

work. Of white drawing papers, the smooth sur¬ 
faced kinds should be selected. For making ink 
drawings, it will be found most satisfactory to use 
the prepared drawing inks, rather than to go to 
the trouble of preparing it oneself from the stick 
India ink. 

For fastening the paper on to the board, common 
one-half-ounce copper tacks are as good, if not 
preferable, to other fastening means. 


CHAPTER II 


DEFINITIONS OF TERMS USED IN GEOMETRICAL 
AND MECHANICAL DRAWING 

1. A Point has position, but not magnitude. 

2. A Line has length, but neither breadth nor 
thickness. 

3. A Surface has length and breadth, but not 
thickness. 

4. A Solid has length, breadth and thickness. 

5. A Plane is a surface which is straight in 
every direction; that is, one which is perfectly 
flat. 

6. Parallel lines are such as are everywhere 
equally distant from each other. Circular lines 
which answer to this condition are also said to be 
concentric. 

7. An Angle is the difference in the direction 
of two lines. If the lines meet, the point of meet¬ 
ing is called the vertex of the angle, and the lines 
ab and ac. Fig. 9, are its sides. 

8. If a straight line meets another so that the 
adjacent angles are equal, each of these angles is 
a right angle, and the two lines are perpendicular 
to each other. Thus the angles acd and dch, Fig. 
10, are right angles, and the lines ab and de are 
perpendicular to each other. A distinction is to 
be made here between the words joerjmidicular 

10 


DEFINITIONS OF TERMS 


11 


and vertical. A vertical line is one which is per¬ 
pendicular to the plane of the earth^s horizon; that 
is, to the surface of still water. 

9. An Obtuse Angle is one which is greater 
than a right angle, as ace, Fig. 10. 

10. An Acute Angle is one which is less than a 
right angle, as ec6, Fig. 10. 

11. It is obvious that the sum of all the angles 
which may be formed about the point c, Fig. 10, 
above the line ah will be equal to the two right 
angles acd and deb. 



Fig. 9.—Angle. Fig. 10.—Illustration for Making 

Clear the Terms Right, Acute 
and Obtuse Angles. 

12. The Complement of an angle is a right angle, 
less the given angle. Thus bee, Fig. 10, is the 
complement of dee. 

13. The Supplement of an angle is two right 
angles less the given angle. Thus bee, Fig. 10, is 
the supplement of ace. 

14. A Circle is a continuous curved line. Fig. 11, 
or the space enclosed by such line, every point of 
which is equally distant from a point within called 
the center. 

15. The distance across a circle, measured 
through the center, is the diameter. The distance 
around the circle is the circumference. The dis- 








12 SELF-TAUGHT MECHANICAL DRAWING 

tance from the center to the circumference is the 
radius. 

16. The ratio between the circumference and 
the diameter, that is, the circumference divided 
by the diameter, is 3.1416. While this is not exact 
(Bradbury’s Geometry states that it has been car¬ 
ried out to two hundred and fifty places of deci¬ 
mals), it is near enough for practical purposes. 
This ratio is frequently represented by the Greek 
letter TT (pi). 

17. A circle is considered as being equally divided 




Fig. 11. -Illustration for ^IG. 12. -Similar Triangles. 

Making Clear the Terms 

Relating to the Circle. 

into three hundred and sixty degrees (360°), each 
degree into sixty minutes (60'), and each minute 
into sixty seconds (60"). 

18. If two diameters cross each other at right 
angles, the circle is divided into four equal parts; 
hence a right angle contains ninety degrees. 

19. An Arc of a circle is any part of its circum¬ 
ference, as ahe, Fig. 11. 

20. A Chord is a straight line joining the ends 
of an arc, as ac. Fig. 11. 

21. Two triangles, as ahe and dec, Fig. 12, hav¬ 
ing like angles are similar triangles. The corre- 





DEFINITIONS OF TERMS 


13 


spending sides of similar triangles have the same 
ratio. Thus if ac were twice as long as dc, ah 
would be twice as long as de, and he would be 
twice.as long as ec. 

22. The sum of the angles of a triangle is equal 
to two right angles. Let ahe, Fig. 13, represent 
any triangle. Extend one side, ac, as shown, and 
make cd parallel with ah. Then the angle dee is 
equal to the angle hac, for their sides have the 
same direction, and the angle hed is equal to the 



Fig. 13.—Illustration for 
Showing that the Sum of 
the Angles in a Triangle 
equals Two Right Angles. 



Fig. 14.—Tangent and Nor 
mal to a Curve. 


angle ahe, for their sides have opposite directions; 
hence the sum of the three angles formed about 
the point c is equal to the sum of the three angles 
of the triangle ahe, and these are equal to two 
right angles (11). 

23. A Tangent is a line which touches another, 
but does not, though extended, cross it. Thus, a, 
b and c, Fig. 14, are tangent lines. A line, d, 
perpendicular to the straight line b, at the point 
of tangency, is called a normal. If one of the 






14 SELF-TAUGHT MECHANICAL DRAWING 

lines, as a, is circular, the normal will pass through 
its center. 

24. A Parallelogram is a figure whose opposite 
sides are parallel, as ah and cd, or eb and fd in 
Fig. 15. The sides may all be of equal length. 



Fig. 15.—Parallelograms. Fig. 16.— Square. 

when the parallelogram is called a square. (See 
Fig. 16.) 

25. Figures having five, six or eight sides are 
called respectively Pentagon, Hexagon and Octagon. 
These, and all figures having more than four sides, 
are called Polygons. If the sides in a polygon are 



Fig. 17.—Regular Polygon. Fig. 18.—Ellipse. 

all of equal length, and all the angles equal, the 
polygon is called a regular %>olygon. (See Fig. 17.) 

26. An Ellipse, Fig. 18, is a continuous curved 
line, or the space enclosed by such line, of such 
shape that the sum of the distances from two 













DEFINITIONS OF TERMS 


15 


points within, as a and h, called the foci (singu¬ 
lar: focus), to any point upon its circumference 
is constant. Thus al plus hi equals a2 plus h2 or 
aS plus bS. 

27. An Involute is a line of such shape (as a in 




Fig. 19.—Involute. Fig. 20.—Cycloid. 


Fig. 19) as might be made by a pencil at the end 
of a string which is unwound from a circle. 

28. A Cycloid is a line of such shape (as a in 
Fig. 20) as might be made 
by a pencil fastened to the 
circumference of a circle 
which is being rolled upon 
a straight line. If the circle 
was being rolled upon the 
convex side of a circular 
line the line traced by the 
pencil would be an epicy¬ 
cloid, If it was being rolled 
upon the concave side of a 
circular line, the line traced 
by the pencil would be a 
hypocycloid. The involute 
and cycloidal curves are used in gear outlines. 

29. A Parabola is a curve which may be ob- 



Fig. 21. Method of Sec¬ 
tioning a Cone to Ob¬ 
tain a Parabola. 




16 


SELF-TAUGHT MECHANICAL DRAWING 


tained by cutting a cone so that the exposed 
sectional surface will he parallel with one of the 
sides of the cone, as shown in Fig. 21. This 
curve, as shown in Fig. 22, is of such shape that 
lines drawn to it from a certain point within, 
called the focus, shown at / in the illustration, 
make the same angle with it as lines drawn from 



the intersection points parallel with the axis ax. 
Thus the line fm makes the same angle with the 
parabola, at the point of intersection, as the line 
ml. Because of this property of the parabola, 
mirrors of this shape are used in headlights of 
locomotives, in search lights, and in many light¬ 
houses; because, if a light be placed at the focus, 
its rays, when reflected from the mirror, will be 
thrown out in parallel lines. 









CHAPTER III 


GEOMETRICAL PROBLEMS 


Prob. Fig. 23. To bisect a line, either curved 
as abc, or straight as ac. —With centers at a and c 
and with a radius somewhat greater than half the 
length of the line, describe the arcs d and e. A 
line passing through the intersections of these arcs 
bisects either line. It will also pass through the 
center of the circle of which the arc abc is a part. 

Prob. 2, Fig. 2Jf. To bisect an angle .—With 



Fig. 23.—Bisecting a Line. Fig. 24.—Bisecting an Angle. 

center at a, and with any convenient radius, de¬ 
scribe the arc be. With centers at b and c, and 
with a radius greater than half the arc, describe 
the arcs d and e. A line from a through the inter¬ 
section of these arcs bisects the angle. 

Prob. 3, Fig. 25. To make an angle equal to a 
given angle. —Let a be the given angle, and let it 
be desired to make an angle equal to it on the line 
dg. With center at a make the arc be, and then 
with center at d make the arc eh with the same 


17 





18 


SELF-TAUGHT MECHANICAL DRAWING 


radius. Then with a radius equal to 6c, and with 
center at h, make the arc /. A line from d through 
the intersection of the arcs gives the required 
angle. 

Proh. Jf, Fig, 26. To erect a perpendicidar at the 
end of a line, ah. —With any convenient center, c, 




Fig. 25.—Making an Angle Equal to a Given Angle. 


and with radius cb, draw a semicircle intersecting 
ab at d. Draw a line from d through c intersect¬ 
ing the semicircle at e. A line from 6 passing 
through e is the required perpendicular. 

Prob. 5, Fig. 27. To drop a perpendicular from 
a point a, to a given line be. —With a as a center. 



Fig. 26.—Erecting a Perpen¬ 
dicular Line. 




/ 


a 




X 


9 


Fig. 27.—Drawing a Perpen¬ 
dicular Line. 


draw an arc intersecting be at d and e. With d 
and e as centers draw the intersecting arcs / and 
g. A line from a through the intersection of 
these arcs is the required perpendicular. If a 
were over one end of the line be the process shown 









GEOMETRICAL PROBLEMS 


19 


in the preceding problem might be reversed by 
drawing a line from a corresponding to de, Fig. 
26, and upon this line drawing a semicircle, when 
its intersection with the base line would give the 
point to which the perpendicular from a should be 
drawn. 

Prob. 6, Fig. 28. To draw a tangent to a circle 
at a given point. —Draw a radius of the circle to 
the required point, and erect a perpendicular to it, 
which will be the required tangent. To find the 
point of tangency of a line to a circle, drop a per- 



Fig. 28.—Drawing a Tangent Fig. 29.—Finding the Center 
to a Circle. of a Circle. 

pendicular to the tangent from the center of the 
circle. 

Prob. 7, Fig. 29. To find the center of a circle .— 
Mark off two arcs as ab and ac upon the circumfer¬ 
ence, and bisect these arcs as in Prob. 1. Where 
these bisecting lines cross each other will be the 
required center. 

Prob. 8, Fig. SO. To draiv a regidar hexagon 
upon a given base, aft.—With a radius equal to the 
length of ab draw the arcs c and d. The intersec¬ 
tion of these arcs will be the center of a circum¬ 
scribing circle upon which the other sides may be 
marked off. 




20 SELF-TAUGHT MECHANICAL DRAWING 

Proh. 9, Fig. 31. To draw a regular octagon in 
a square. —Draw the diagonals of the square, ad 
and he, and with a radius equal to half of a diago¬ 
nal, and with centers at a, h, c and d, draw the 
arcs e, /, g and h. The intersections of these arcs 



Fig. 30.—Drawing a Regular Fig. 31.—Drawing a Regular 
Hexagon. Octagon. 

with the sides of the square give the corners of 
the required octagon. 

Proh. 10, Fig. 32. To draw a circle about a tri¬ 
angle, as ahc. —Bisect any two of the sides as in 
Proh. 1. Where the bisecting lines cross each 



Fig. 32.—Drawing a Circle Fig. 33.—Inscribing a Circle 
about a Triangle. in a Triangle. 


other will be the center of the required circle. In 
a similar manner a center may be found from 
which to draw a circle through any three given 
points, the given points in this case being the cor¬ 
ners of the triangle. 









GEOMETRICAL PROBLEMS 


21 


Prob, 11, Fig. 33. To draw a circle within a 
given triangle, as abc. —Bisect any two of the angles 
as in Prob. 2. Where the bisecting lines cross, will 
be the center of the required circle. In a similar 
manner a center may be found from which to draw 
a circle tangent to any three given straight lines. ' 
Prob. 12, Fig. 3k. To find the foci of an ellipse .— 
Draw the long and the short diameters of the 
ellipse, ab and cd, and with a radius equal to half 
of the long diameter, and with a center at c or cZ 


c 




Fig. 34.— Finding the Foci of Fig. 35.— Simplified Method 
an Ellipse. of Drawing an Ellipse. 

draw the arcs e and/. Where these arcs intersect 
the long diameter will be the required foci. 

Prob. 13, Fig. 35. To draw an ellipse with a 
pencil and thread. —Having found the foci of the 
ellipse, stick a pin firmly into each focus, and loop¬ 
ing a thread around them, allow it to be slack 
enough so that the pencil will draw it out to the 

end of the short diameter. The thread will then 

) 

guide the pencil so that it will draw an ellipse. A 
groove should be cut around the pencil lead to pre¬ 
vent the thread from slipping off. 

Prob. Ik, Fig. 36. To draw an ellipse with a 
trammel.— hsiY out the long and the short diame¬ 
ters of the ellipse, ab and cd, and on a strip of 
paper. A, mark off 1-3 equal to half of the long diam- 







22 SELF-TAUGHT MECHANICAL DRAWING 

eter, and 2-3 equal to half of the short diameter., 
Then, keeping point 1 on the short diameter, and 
point 2 on the long diameter, mark off any desired 
number of points at 3. . A curved line passing 
through these points will be the required ellipse. 
The ellipsograph, an instrument for drawing el¬ 
lipses, is made on this principle, points at 1 and 
2 traveling in grooves which coincide with ab 
and cd. 

Prob. 15, Fig. 37. To draw an ellipse by tangent 
lines.—ab equal to one-half of the long di- 



Fig. 36.—Another Method of Fig. 37.—Drawing an Ellipse 
Drawing an Ellipse. by Tangents. 

ameter of the required ellipse, and be equal to one- 
half its short diameter. Divide ab and be into 
the same number of equal parts, and, numbering 
them as indicated, connect 1 and T, 2 and 2' and 
so forth. A curved line starting at a, tangent to 
these lines, and ending at e, is one-quarter of the 
required ellipse. 

Prob. 16, Fig. 38. To draw an approximate el¬ 
lipse with eompasses, using four eenters. —Lay out 
the long diameter ab, and the short diameter ed, 
crossing each other centrally at o. From b meas¬ 
ure off be equal to eo, one-half of the short diam¬ 
eter. The length ae will then be the radius gh 
for forming the part hk of the ellipse. From e 







GEOMETRICAL PROBLEMS 23 

mark off the point/, making ef equal to one half 
of oe. The point / will be the center, and/6 the 
radius for forming the end of the ellipse. Lines 
drawn from the centers g through the points / de¬ 
termine the points at which the different curves 
meet. This method is not considered applicable 
when the short diameter is less than two-thirds of 
the long diameter. 



d 

Fig. 38.—Drawing an Approximate Ellipse by Four 
, Circular Arcs. 


Prob. 1 7, Figs. 39 and 39a. To draiu an approx¬ 
imate ellipse tvith compasses, using eight centers .— 
Lay out the long diameter ah, and the short diam¬ 
eter cd crossing each other centrally at /. Con¬ 
struct the parallelogram aecf, and draw the diago¬ 
nal ac. From e draw a line at right angles to ac, 
crossing the long diameter at h, and meeting the 
short diameter, extended, at g. Point g is the center 
from which to strike the sides of the ellipse, and 






24 SELF-TAUGHT MECHANICAL DRAWING 

h will be the center, subject to certain modifica¬ 
tions for narrow ellipses, from which to strike the 
ends of the ellipse. To get the radius of the third 
curve for connecting the side and end curves, lay 
off a base line ah, Fig. 39A, of any convenient 
length, and divide it into five equal parts by the 
points 1, 2, 3 and U. At one end of the line erect 
the perpendicular ac, equal to the end radius ah, 
and at the other end erect the perpendicular hd 
equal to the side radius eg. Connect the ends of 
these perpendiculars by the line cd, and at point 
2 erect a perpendicular, meeting cd at e. The 
length e2 will be the desired third radius. With 
the compasses set to this radius, find a center i 
from which a curve can be struck which will be 
just tangent to the side and end curves. From 
other centers similarly located the remainder of 
the ellipse is drawn. Lines drawn from i through 
h, and from g through i determine the meeting 
points of the different curves. 

For narrow ellipses the length of the end radius, 
ah, should be increased as follows: For an ellipse 
having its breadth equal to one-half of its length, 
make ah one-eighth longer. For an ellipse having 
its breadth one-third of its length, make ah one- 
fourth longer. For an ellipse having its breadth 
equal one-quarter of its length, make ah one-half 
longer. For intermediate breadths lengthen ah 
proportionately. With this modification of the 
length of the end radius, this method gives curves 
which blend well together so as to satisfy the eye, 
and gives a figure which conforms quite closely to 
the actual outlines of an ellipse. 




GEOMETRICAL PROBLEMS 


c 



<1 



Fig. 39a. 

Figs. 39 and 39a.—Drawing an Approximate Ellipse by 

Eight Circular Arcs. 


25 














26 SELF-TAUGHT MECHANICAL DRAWING 

Proh. 18, Fig. JfO. To draw a regular polygon of 
any number of sides cm a given base, ab. —Extend ab 
as shown, and on it with one end as a center and 
a radius equal to the length of the given side, draw 
a semicircle. Divide this semicircle into as many 
equal spaces as there are to be sides to the polygon. 
A line from b to the second space, reckoning from 
where the semicircle meets the extension of ab, 
will be a second side of the required polygon. 
Lines are then drawn from b through the remain¬ 
ing divisions of the semicircle, and the remaining 




P’'iG. 40.— Drawing a Regular Fig. 41. —Drawing a Spiral 
Pentagon. about a Square. 

sides of the polygon are marked off upon them as 
indicated. If the polygon is to have many sides, 
as an additional precaution against error, bisect ab 
and b2, thus getting the center of a circumscribing 
circle upon which the remaining sides may be 
marked off. 

Prob. 19, Fig. J^l. To draw a spiral about a 
square. ~hsLy out a square, 1-2-3-Jf, having the 
length of each side equal to one-quarter of the de¬ 
sired distance between the successive convolutions 
of the spiral, and extend each side in one direction 

t 

as shown. With a center at 2, and with a radius 
1-2 draw a quarter of a circle. With a center at 3 



GEOMETRICAL PROBLEMS 


27 


draw another quarter of a circle, continuing the 
first one, and so continue with successive corners 
of the square for centers. 

Fig. 42 shows how, by similarly extending one 
end of each side, a spiral may be drawn about a 
regular polygon of any number of sides. A curve 
so formed determines the shape of the teeth of 
sprocket wheels. 

Prob, 20, Fig. US. To draw an involute .—Upon 
the circumference of the given circle mark off any 



Fig. 42. —Drawing a Spiral 
about a Regular Polygon. 



lute. 


number of equally distant points, as 0-1-2-S, etc., 
and draw lines tangent to the circle at these points, 
beginning at point 1. Then with the compasses 
set the same as for marking off the spaces on the 
circle, mark off one space on line 1, two spaces on 
line 2, three spaces on line 3, and so forth. A 
curved line starting at 0 and passing through these 
points will be the required involute. This curve 
is used for the shape of the teeth of involute gears. 

Prob. 21, Fig. UU> To draw a cycloid .—Upon the 
base line ab mark off any number of equally dis¬ 
tant points, as G-1-2-3, etc., the distance between 



28 SELF-TAUGHT MECHANICAL DRAWING 


them being made, for convenience sake, about one- 
sixth of half the circumference of the generating 
circle. Beginning at 1 erect perpendiculars from 
these points, and with centers on these lines draw 
arcs of circles tangent to the base line to represent 



Fig. 44.—Drawing a Cy¬ 
cloid. 


Fig. 45.—Drawing an Epicy¬ 
cloid. 


successive positions of the generating circle as it 
is rolled along. With the compasses set as for 
spacing off the base line, mark off one space on the 
arc which starts from point 1, two spaces on arc 
2, three spaces on arc 3, and so forth. A curved 

line starting at 0 and pass¬ 
ing through the points 
thus obtained will be the 
required cycloid. 

An epicycloid. Fig. 45, 
or a hypocycloid. Fig. 46, 
is formed in precisely the 
same way, excepting that 
as the base line, a6, is an arc of a circle, the center 
lines from points 1-2-3, etc., are made radial. 

These three cycloidal curves are used for the 
shape of the teeth of epicycloidal gears, sometimes 
called simply cycloidal gears. 



Fig. 46.- 


-Drawing a Hypo- 
cycloid. 














GEOMETRICAL PROBLEMS 


29 


Proh. 22, Fig, U7. To draw a parabola by means 
of intersecting lines. —Draw the axis ax, and on it 
mark the focus / and the vertex v, and at right 
angles to it draw the line be at a distance from v 
equal to the distance of v from/. Across the axis, 
and at fight angles to it, draw a number of lines, 
1, 2, 3, U, 5, 6. Then with radius al, and with 
center at the focus /, draw arcs intersecting line 
,1; with radius a2, and with center again on/draw 
arcs intersecting line 2, and so on. A curved line 



passing through these intersections will be a para¬ 
bola. It will be seen from this method of drawing 
a parabola that any point on it is equally distant 
from the focus, and from the line be, called the 
directrix. 

Prob. 23, Fig. U8. To draw a parabola with a 
pencil and string. —Lay out the axis, the focus, the 
vertex and the directrix as before. Attach one 
end of a thread to the focus, / by means of a pin, 
and attach the other end of the thread to the 
square shown at d, having the thread of such 















30 SELF-TAUGHT MECHANICAL DRAWING 

length that when the inner edge of the square is 
on the axis, ax, the thread if drawn down with 
a pencil will just reach to the vertex, v. Now 
slide the square along be in the direction of the 



Fig. 48.—Simplified Method of Drawing a Parabola. 

arrow, keeping the pencil against the square; the 
thread will cause the pencil to move along so as to 
describe a parabola as shown. 

Prob. 2Jf, Fig. U9. To draw a parabola of a given 



Fig. 49.—Another Method of Drawing a Parabola. 

breadth of opening, ab, and of a given depth, cd .— 
Draw ef parallel with ab, and draw ae and bf paral¬ 
lel with cd, having ac and be equal Space off de 

















GEOMETRICAL PROBLEMS 


31 


and df into any number of equal parts, and also 
space off ea and/6 into the same number of equal 
parts, as shown. From d draw lines to the di¬ 
visions on ea and/6, and from i, 2, 3 and ^ on de 
and df draw perpendicular lines to intersect the 
lines drawn from dio 1, 2, 3 and U on lines ea and 
/6. A curved line passing through these inter¬ 
sections will be the required parabola. 

Prob. 25, Fig. 50. To find the focus of a para¬ 
bola. —Let abed be the given parabola, ef being its 



Fig. 50.—Finding the Focus of a Parabola. 


axis. Across the parabola at its vertex, v, draw 
the line ij at right angles to the axis. From any 
point, g, on the parabola, draw the line gh parallel 
to the axis. With center at g find a radius, by 
trial, which will cut the axis as much inside the 
vertex, v, as it cuts the line gh beyond the line ij. 
The intersection at x will be the required focus. 






CHAPTER IV 


PROJECTION 

Mode of Representing Objects.—In mechanical 
drawing, machines, or parts of machines, are rep¬ 
resented by views, generally three, in which per¬ 
spective is ignored, and which show the object in 
different positions at right angles to each other. 
The mode of representing these views, and their 
positions with regard to one another, which expe¬ 
rience has shown to be most convenient is perhaps 
best shown by means of the familiar cardboard 
illustration. Let ahcdefgh, Fig. 51, represent a 
piece of cardboard, which we will suppose to be 
transparent, creased on the dotted lines to permit 
of the outer portions being turned back. Let us 
now suppose that we have a prism shaped as shown 
at C, and of the length shown at A. If the prism 
is stood upright with its broad side facing the ob¬ 
server, and the cardboard, being blank, is held up 
in front of it, the prism will appear, if all its lines 
are brought perpendicularly forward to the card¬ 
board, as it is shown at A, lines on the prism 
which would be hidden by its body, as the further 
corner, being dotted. If section C of the cardboard 
is now turned backward through an angle of 90 
degrees over the top of the prism we would get the 
view shown in that part, all lines being brought 

32 


PROJECTION 


33 


perpendicularly forward from the prism to the 
cardboard as before. Likewise if part D of the 
cardboard were turned backward through an angle 
of 90 degrees, and the lines of the prism were 
brought perpendicularly forward onto it, we would 
get the view shown in that part. The view shown 
at A is called the elevation, that shown at C is 
called the plan, and that shown at D is called the 
side view. Occasionally a piece is so shaped, or 



has so much of detail to it as to make another side 
view desirable; such a view would be placed at B. 
In many other cases, as in the case of the prism 
here shown, the plan and elevation views alone 
will fully show the object. 

The production of these views from one another 
is called projection; and by the use of connecting 
lines, and also at times of temporary construction 
views, objects may be shown at any desired angle, 
irregular or curved lines may be traced, and sur¬ 
faces may be developed. 



















34 


SELF-TAUGHT MECHANICAL DRAWING 


An Upright Prism.—Fig. 52 shows a prism in its 
simplest position. A moment’s examination will 
show that the elevation cannot be drawn directly, 
as the distance apart of the vertical lines which 
represent the corners of the prism, cannot be deter¬ 
mined without other aid; hence it is necessary to 
draw the plan view first. Horizontal lines having 
been made to give the height of the prism in the 
elevation, the vertical lines may then be drawn in 



Fig. 52.—Projections of 
Prism. 



Fig. 53.—Projections of 
Tilted Prism. 


from the plan, as indicated by the vertical dotted 
line. 

The Prism Inclined at One Angle.—Fig. 53 shows 
the prism inclined to the right. A brief exami¬ 
nation of these views will show that none of them 
can be drawn directly, as the distance apart of 
the vertical lines in the elevation and side views 
is not known, and the lines of the plan view are 
foreshortened; but the views can be developed 
from Fig. 52. It is evident that as the prism is 
tipped, the elevation view will remain unchanged. 



















PROJECTION 


35 


hence the first step will be to reproduce that view 
inclined at the desired angle. As the prism is 
tipped it is also evident that all points in the plan 
view of Fig. 52 will move in horizontal lines to the 
right, hence horizontal lines are drawn from these 
points through the position which the plan will 
occupy in Fig. 53. The intersection of these lines 
with vertical lines from the corresponding points 
in the elevation will determine the position of 
each point in the plan. The points so determined 
one by one being then connected by straight lines, 
gives the plan view as shown. To make the side 
view, horizontal lines are first drawn from the 
various points of the prism as seen in the eleva¬ 
tion through the position which the side view will 
occupy. Then, bearing in mind that each point 
of the prism in the side view will be as much to 
the left of the vertical line ab as the same point 
in the plan is below the line cd, the position of 
each point on the horizontal lines is marked off 
from ab. 

The Prism Inclined at Two Angles.—Fig. 54 shows 
the prism tipped forward after having been tipped 
to the right as shown in Fig. 53. An examination 
of these views will show that not only can they 
not be drawn directly, but they cannot be devel¬ 
oped from Fig. 52. They may, however, be de¬ 
veloped from Fig. 53. It is evident that as the 
prism is tipped forward, the side view of Fig. 53 
will remain unchanged; hence the first step will 
be to reproduce that view inclined at the desired 
angle. Next, horizontal lines are drawn from the 
corners of the prism as seen in this view through 


36 SELF-TAUGHT MECHANICAL DRAWING 

the place which the elevation is to occupy, and the 
perpendicular line gh is drawn. It is evident that 
as the prism is tipped forward, the different points 
of it as seen in the elevation of Fig. 53 do not 
move any to the right or left, but forward only. 
Hence, the distance of the corners of the prism 
from the line ef may be taken by the compasses 
and marked off from the line gh upon the proper 
horizontal line. The new position of all of the 

corners having thus been 
determined, the con¬ 
necting straight lines 
are drawn, giving the 
elevation as shown in 
Fig, 54. Vertical lines 
are then drawn from the 
different points of the 
prism, as seen in this 
view, through the posi¬ 
tion which the plan is 
to occupy, and the exact 
position of each point 
upon these lines is 
marked off from mn at the same distance which 
it is from the line jk in the side view. 

An Upright Rectangular Prism.—The upright 
rectangular prism shown in Fig. 55 is, of course, 
drawn in the same way as was the prism shown 
in Fig. 52. 

The Prism of Fig. 55 Tipped Forward on One Edge. 
—It is evident that if the prism were to be tipped 
on its edge in the direction of the arrow No. 1, the 
result would be the same as though it had been 



Fig. 54.—Projections of Prism 
Tilted in Two Directions. 





projection 


87 


tipped first to the right, and then directly forward, 
as was done to produce Fig. 54; but as those 
angles are not given, the method employed in that 
case is not readily available. 

Fig. 56 shows the prism tipped to its new po¬ 
sition, and shows, also, the method employed to 
produce the views. Draw the line cd at the same 




Fig. 55.—Upright Rectan- Fig. 56.—Rectangular Prism 
gular Prism. Tipped Forward. 


angle to the horizontal as the edge ah of the prism 
in Fig. 55, and make ef at right angles to it. Upon 
these lines draw the temporary side view of the 
prism. A, tipped at the desired angle. With the 
aid of this view the plan view is readily drawn. 
Vertical lines are then drawn from the various 
points of the plan view through the place which 
the elevation is to occupy, and the exact location 
of each point is marked off on these lines at the 













38 


SELF-TAUGHT MECHANICAL DRAWING 


same height above the base line gh that it is above 
the line ef in the temporary side view, A, The 
permanent side view is then developed from 
the plan and elevation in the same way as was the 
side view of Fig. 53. 

Let it now be required to tip the prism of Fig. 55 
forivard on one corner in the direction of arroiv 


No. 2. 


It will be seen that tipping it in this direction 



Fig. 57.—Rectangular Prism Tipped in Two Directions. 

will cause a foreshortening of all of the lines in the 
plan, hence the use of a single tem.porary view 
such as was used in Fig. 56 will not solve the 
problem; but it may be solved by the use of two 
temporary views as shown in Fig. 57. Draw the 
line ab in the direction in which the prism is to be 
tipped, and the line cd at right angles to it. At A 
reproduce the plan view of Fig. 55, and at B draw 




PUO.JECTION 


39 


a side view of the prism as it would appear if A 
were viewed in the direction of the arrow, but in¬ 
clined to cd at the required angle. The intersec¬ 
tion of lines drawn from the corners of A, parallel 
with ab, with lines drawn from the same corners 
of B, parallel with cd, will give their location 




in the permanent plan view. This view being 
finished, the elevation and the permanent side 
views are drawn in the same way as were those 
of Fig. 56. 

Let a cube be set on one corner so that a diagonal 
of it shall be horizontal; required to show the angle 
which the edges that meet at that forward corner 
make with a plane perpendicular to the diagonal, 
the angle which the sides that have corners coming 
together at the same point make with the plane, and- 














40 SELF-TAUGHT MECHANICAL DRAWING 

also the amount of foreshortening of the lines which 
will he caused. 

In Fig. 58, A shows a face view of the cube set 
on edge, B shows a side view of the same, and C 
shows B inclined until the diagonal ke becomes 
horizontal. The length of he being laid out on the 
center line, the position of the other corners is ob¬ 
tained as indicated by the arcs a, 6, c and d. The 
angle geh is the required angle which the ed^es 

which meet at e make 
with a plane perpendic¬ 
ular to ek, of which fg is 
an edge view; the angle 
fej is the angle which 
the sides having corners 
meeting at e make with 
the plane. Z) is a face 
view of C, and any of its 
lines, when compared 
with any of the lines of 
A, will show the fore¬ 
shortening caused by the 
cube being put into this 
position. 

The Surface Develop¬ 
ment of a Cone.—Let A 
and B, Fig. 59, be the 
plan and elevation views 
of a cone. With a radius 
equal to ah, and with a 
center at c, draw the arc def making it equal in 
length to the circumference of the base of the cone, 
as shown at A. This may be most conveniently 



Fig. 59.—Development of 
a Cone. 












PROJECTION 


41 


done by spacing it off. Draw the lines cd and c/, 
and the figure C thus formed will be the required 
surface development. 

The Surface Development of a Pyramid Having 
Its Top Cut Off Obliquely.—In Fig. 60, A, B and C 
show, respectively, the plan, elevation, and side 



Fig. 60.—Development of a Frustum of a Pyramid. 

views of the pyramid, the top of which is cut off by 
the plane ab. These views may be made by the 
principles already explained, as may also the view 
at D, which shows the pyramid as though B were 
viewed in the direction of the connecting dotted 
line, which is at right angles to a6, thus showing 
the shape of the section exposed by cutting off the 
top. 
















42 SELF-TAUGHT MECHANICAL DRAWING 

To get the surface development, take a radius 
equal to the length of one edge of the pyramid as 
shown at cd in the elevation, this being the only 
one which shows at full length, the others being 
more or less foreshortened, and with a center at e 
in view E, draw an arc of a circle upon which the 
sides of the base are to be marked off. These 
points are connected with one another and with e; 
this gives the shape of the surface of the whole 
pyramid. Upon the lines connecting the points 
with e, asef, e2 and eS, the lengths of the different 
edges of the cut off pyramid are marked off. As 
the edge which is seen at the left in the elevation 
shows full length, its length, dl, may be taken di¬ 
rectly and marked off on the line el. As the other 
edges are seen foreshortened, their lengths cannot 
be taken directly, but by horizontally transferring 
the upper end of each edge to the line cd, their 
actual lengths d2 and dS may be obtained and then 
marked off on the lines e2 and eS. The points so 
obtained being connected, and the outer half sec¬ 
tions being finished, gives the required surface 
development. 

If the cone shown in Fig. 59 were to have its top 
cut off obliquely, the views of it corresponding to 
A, B, C and D, Fig. 60, and its surface develop¬ 
ment, would be obtained by dividing off its base, 
as seen in the plan, into any number of sides, and 
then proceeding as though it were a pyramid of 
that number of sides, until the points correspond¬ 
ing to those of Fig. 60 had been located, but then 
connecting them with curved lines instead of 
straight lines. 


PROJECTION 


43 


Intersecting Cylinders, Fig. 61—Required the 
line of the intersection, the surface development of the 
branch, and the shape which the end of the branch 
timdd appear to have as seen in the view at the 
right 

First draw the elevation, A, in outline, and as 
much of the end view, B, as can be directly drawn. 



Fig. 61.—Intersecting Cylinders. 

Opposite the end of the branch in each of these 
views, and in line with it, draw circles of the same 
diameter as the branch, and space off the semi¬ 
circumference nearest to it into a number of 
equal parts, the same number in both cases. From 
the points so obtained draw lines parallel with the 
center line of the branch, as shown. From the 
points where these lines in the view B meet the 

























44 SELF-TAUGHT MECHANICAL DRAWING 

circle representing the end of the large cylinder, 
draw horizontal lines intersecting the lines drawn 
from C, These intersections will be points through 
which the line of the intersection of the cylinders 
is to be drawn. From the points where the lines 
drawn from C cross the end of the branch, draw 
horizontal lines intersecting those drawn from D, 
These intersections will be points through which 
the line representing the end of the branch is to 
be drawn. 

To get the surface development of the branch, 
first draw the line ah, in E, having it in line with 
the end of the branch. Make this line equal in 
length to the circumference of the branch, spacing 
it off equally each way from the center line OX into 
the same number of spaces as the semi-circumfer¬ 
ence of C was divided into. From these points 
draw lines parallel with OX, and from the points 
in the intersection of the two cylinders, previously 
obtained, draw lines parallel with ah, intersecting 
these lines. These intersections will be points 
through which a curved line is to be drawn, thus 
giving the completed surface development of the 
branch. 

In drawing these curved lines through the points 
of intersection, the irregular curves mentioned in 
the early part of the chapter on instruments and 
materials are used. 

Intersecting Cylinder and Frustum of Cone, Fig. 

62 .—Required line of intersection and surface de¬ 
velopment of branch, as before. 

Draw the elevation, A, in outline, continuing the 
sides of the conical branch either way until they 


PROJECTION 


45 


meet at their vertex, a, on the one hand, and to 
any convenient points, c and d, on the other. In a 
similar manner draw as much of the end view, B, 



Fig. 62.—Intersecting Cylinder and Cone. 


as can be made directly. With centers at a and at 
h, and with any convenient radius, draw the arcs 
cd and c/, intersecting* the extended sides of the 
conical branch. Then, with centers at the inter- 















46 


SELF-TAUGHT MECHANICAL DRAWING 


section of these arcs with the center line of the 
branch, draw the half ci cries shown, tangent to 
the extended sides of the branch, and space them 
off into a number of equal parts, the same number 
- in each case. From these points draw lines to the 
vertices a and h. From the points where these 
lines in the end view, B, intersect the circle repre¬ 
senting the end of cylinder, draw horizontal lines 
to the elevation. A, intersecting the lines drawn 
from the vertex a to the half-circle cd. The inter¬ 
sections will be points through which the line rep¬ 
resenting the intersection of the cylinder and its 
conical branch is to be drawn. The shape of the 
end of the branch as seen in the end view, B, is 
now obtained in the same manner as in the case of 
the intersecting cylinders. From the points where 
the lines drawn from the vertex, a, of the side ele¬ 
vation A, to the half-circle at cd, cross the end of 
the branch, draw horizontal lines intersecting the 
lines drawn from the vertex h. These intersec-, 
tions will give points through which the line rep¬ 
resenting the end of the branch in view B is to 
be drawn. 

To get the development of the branch as shown 
at F take a radius equal to the distance from the 
apex a to the end of the branch as seen in the side 
elevation. A, and with a center at g draw an arc 
hi, making the length of the arc equal to the cir¬ 
cumference of the end of the branch as shown at 
E, spacing equally each way from the center line 
gj, the length and number of the spaces each way 
being the same as those obtained in spacing off the 
semicircle at E, Through these points draw lines 


PROJECTION 


47 


radiating from g, as shown. On these lines dis¬ 
tances are marked off from the arc hi through 
which the irregular curved line is drawn which 
gives the development of the branch. The lengths 
at the middle and at the extremities may, of course, 
be taken directly from the elevation A, the length 
kl being the length on the center line, and the 
length mn being the length at the extremities. 
The other lengths, being foreshortened, as seen in 
the elevation A, cannot be taken directly, but are 
obtained by transferring the points to either kl or 
mn as shown by the dotted lines, as was done in 
the case of the pyramid. Fig. 60. 

To Draw a Helix.—A helix is a line of such shape 
as would be made by winding a thread around a 



Fig. 63.—Drawing a Helix. 



Fig. 64.—The Helix as 
it Appears in a Screw 
Thread. 


cylinder^ and having it advance lengthwise on the 
cylinder at a uniform rate as it is wound around 
it. In Fig. 63 we have the side and end views of 
a cylinder upon which it is desired to draw a helix, 
which shall advance from a to 6 in making a half 
turn around it. Divide the space from a to 6 into 
any number of equal parts, and at the points so 
obtained erect perpendicular lines. Divide the 

























48 


SELF-TAUGHT MECHANICAL DRAWING 


semi-circumference of the end view of the cylinder, 
toward the side view, into the same number of 
equal parts, and from these points draw horizontal 
lines to meet the perpendiculars previously erected. 
Where these lines meet will be points through 
which the helix is to be drawn. 

The outlines of a screw thread are helices. Fig. 
64 shows a double threaded Acme standard, or 29 
degree threaded screw, the outline of which, on 
its outside diameter, is the helix of Fig. 63. 

Isometric Projection.—If a cube is tipped over on 
one corner, so that the diagonal of it is horizontal 
as shown at Z>, Fig. 58, and also in Fig. 65, the 



Fig. 65.— Principle of Fig. 66.— An Example of 

Isometric Projection. Isometric Projection. 

lines of it will all appear of equal length. Draw¬ 
ings made on this principle, as Fig. 66, are called 
isometric drawings. Vertical' lines remain ver¬ 
tical. Horizontal lines become inclined to the 
horizontal of the paper at an angle of 30 degrees. 
Circles appear as ellipses, which may be drawn as 
shown in the upper square of Fig. 65. From the 
ends of the “ short’^ diagonals, lines are drawn to 
the middle of the opposite sides. Where these 
lines cross the ‘dong” diagonals are located the 
centers from which the ends of the ellipse may 









PROJECTION 


49 


be drawn. The ends of the short diagonals will 
be centers from which to draw the sides of the 
ellipse. 

Irregular curves may be drawn as indicated in 
Figs. 67 and 68. The figure 2 there shown is first 
drawn in the desired position in a naturally shaped 
square, which is then divided off by equally spaced 
lines into smaller squares. The isometric square 
is then similarly divided off, and the figure is 




Figs. 67 and 68. —Method of Transferring Irregular Lines in 

Isometric Projection. 

made to pass through the corresponding inter¬ 
sections. 

Isometric drawings differ from perspective draw¬ 
ings in that receding lines remain parallel, instead 
of converging to a vanishing point. They may be 
measured the same as ordinary drawings in any 
one of the three directions indicated by the lines 
of the cube. The foreshortening of the lines caused 
by tipping the cube into this position is generally 
ignored. If an isometric drawing is to be shown 
in connection with ordinary views, however, it 
should be made on a scale of about 8-10 of an inch 
to the inch, otherwise it would appear too large. 


CHAPTER V 


. WORKING DRAWINGS 

As the object of working drawings is to convey 
to the workman a clear idea of the appearance and 
construction of the piece to be made, and as the 
whole ''science’^ of mechanical drawing has been 
developed primarily for the purpose of conveying 
the ideas and thoughts of the designer and drafts¬ 
man to the men who carry out these ideas in wood 
and metal, the subject of working drawings is of 
supreme importance to all mechanics. A working 
drawing should be as complete as possible, so com¬ 
plete, in fact, that when it has once passed out of 
the draftsman’s hand into the shop, no further 
questions will be necessary. In order to accom¬ 
plish this, all necessary information, of whatever 
kind, should be included, and, if required, short 
notes and directions may be written on the draw¬ 
ing to prevent eventual misunderstandings. 

The number of views necessary to, properly rep¬ 
resent an object must be left for the draftsman’s 
judgment to determine. Usually two views are 
sufficient, when the object is simple, but when at 
all complicated, three or more views will be found 
necessary. Cylindrical pieces can often be ade¬ 
quately represented by a single view, on which the 
various diametral and length dimensions are given. 

50 


WORKING DRAWINGS 


51 


While it is customary to put the plan view of an 
object above the elevation, it frequently becomes 
necessary, in order to present the objects shown in 
as clear a manner as possible, to deviate from this 
rule. A case of this kind is shown in Fig. 69, 
where the shaft hanger illustrated has been se¬ 
lected as an example of the methods employed in 
working drawings. 

An examination of the hanger will show that if 
the plan were placed above the elevation, and if it 
were represented according to the methods already 
explained, the box and the yoke with its adjusting 
screws and check-nuts would have to be shown 
mostly by dotted lines. Such a multiplicity of 
dotted lines would tend to confusion; hence the 
object in view, that of presenting the hanger in as 
clear a manner as possible, is best accomplished in 
a case like this by having the plan underneath the 
elevation, and letting it be a bottom view instead 
of a top view. 

In designing a machine detail of this kind, the 
starting point would of necessity be the shaft 
itself, and the first step would be to design the 
box; next would come the yoke, and lastly, the 
frame. Much of the preliminary \vork may fre¬ 
quently be done on scrap paper; having determined 
the size and proper proportions of the various 
parts, the position which the different views will 
occupy in the finished drawing is easily ascer¬ 
tained. The center lines are then laid out as 
shown, and the drawing built up about these lines 
as a base. 

When a drawing is for temporary use only, and 


52 


SELF-TAUGHT MECHANICAL DRAWING 


the mechanism represented on it of a simple nature, 
the assembly drawing, corresponding to the three 
views in Fig. 69, will answer all purposes, the di¬ 
mensions being given directly on this drawing. In 



some cases only the most important dimensions 
would be given, those of secondary consequence 
being left for the workman to be obtained by 
“scaling’^ the drawing. This procedure, however, 
is possible only when the drawing is made care¬ 
fully to scale, and is not one that should be en- 




































































































WORKING DRAWINGS 


53 


couraged. In general, a drawing should be- so di¬ 
mensioned that it can be worked to without the 
workman obtaining any measurements by “scal¬ 
ing’' the drawing. 

In most cases it is not possible to show the de¬ 
tails of a mechanism clearly enough in an assembly 
drawing; for if the device shown is more or less 
complicated, a hopeless confusion results from the 
attempt to put in all the lines necessary to fully 
show all the details; neither would it be possible, 
for the same reason, to give more than the princi- 



Fig. 70.—Example of Working Drawing. 


pal dimensions. In such cases it is, therefore, cus¬ 
tomary, after the assembly drawing has been com¬ 
pleted, and the proper sizes and proportions of the 
various parts of the mechanism thus ascertained, 
to make a separate drawing of each detail, either 
on the same sheet of paper, or on separate sheets. 
This permits the parts of the mechanism to be 
clearly and completely shown and fully dimen¬ 
sioned Figs. 70 and 71 show two pieces of the 
hanger in Fig. 69 detailed in this manner. These 
detail drawings give all the required informa¬ 
tion for the making of the pieces, and the assembly 


































54 


SELF-TAUGHT MECHANICAL DRAWING 


drawing merely shows, in a general way, how the 
parts are to be assembled when completed. 

In the case of jig and fixture drawings, it is the 
practice in a great many large drafting-rooms to 
show assembled views only, and to put all dimen¬ 



sions directly on the assembly drawing; the argu¬ 
ment advanced in favor of this practice is that ex¬ 
perienced pattern-makers and tool-makers, who are, 
as a rule, the only mechanics who will work on 
the making of these tools, will find no difficulty in 
reading the assembly drawing; besides, it is said. 
















































































WORKING DRAWINGS 


55 


as a drawing of this kind is, in most cases, used 
but once, it would be waste of time to have the 
draftsman detail the different parts of the tool. 

While these arguments are undoubtedly true in 
the case of very simple jigs and fixtures, there can 
be little doubt that in the case of more complicated 
ones, the comparatively short time required by the 
draftsman to make detail drawings will be saved 
many times over in the shop; for the pattern¬ 
maker and tool-maker will not have to spend, in 
the total, a number of hours puzzling over the draw¬ 
ing, and even then being liable to make a mistake. 

In making drawings, it is always a rule to work 
from the center lines, when the outline of the 
piece is such that it has a definite center line. 
Dimensions in either direction from the center 
line can be best marked off with the compasses. 
This insures a symmetrical appearance to the fin¬ 
ished drawing, such as might not be secured if the 
dimensions are set off on either side of the center 
line from the rule, it always being easy to then 
introduce small errors which show plainly in the 
finished work. If the piece is of such shape as to 
have no center line, some one principal line may 
be selected, one in each direction in each view, 
and the remaining points and lines may be located 
from these lines. 

The various styles of lines ordinarily used in 
working drawings are shown in Fig. 72. The 
regular "‘fulT^ line A A is used for the outlines of 
objects, and when drawn rather “fine,for cross- 
hatching or cross-sectioning. The heavy shade 
line BB is used to represent lines assumed to sepa- 


56 SELF-TAUGHT MECHANICAL DRAWING 


rate the light surfaces of an object from the dark, 
as will be explained in the following. The dotted 
line CC, as has already been explained in the pre¬ 
vious chapter, is used to represent lines obscured 
or hidden from view. The line DD, called a 
‘‘dash’’ line, is used by a great many draftsmen 
for dimension lines. Finally, the line EE, the 
“dash and dot,” or, simply, the “dash-dotted” 


A- 

• 






A 

B 

o- 







-c 

fN 







n 

E 







2 


Fig. 72.—Styles of Lines Used on Working Drawings. 

line, is used in common practice for center lines, 
to indicate sections, etc. This line is also com¬ 
monly used for construction lines, in laying out 
mechanical movements. 

The dimension lines may be made either fine 
full lines or “dash” lines, the dashes being about 
§ inch long. A space is left open for the figures 
giving the dimension. The witness points or ar¬ 
row heads, showing the termination of the dimen¬ 
sion, are made free hand. Many draftsmen draw 
the extension and dimension lines in red ink, the 
arrow heads, however, still being made black. It 
is well to avoid, as far as possible, having the 






















WORKING DRAWINGS 


57 


dimension lines cross each other, as such crossing 
tends to confusion; the difficulty can usually be 
avoided by having at least one set of dimensions 
placed outside or between the views, the larger di¬ 
mensions being placed farther frorh the outline of 
the object than the shorter ones, to avoid having 
the extension lines of the latter cross the dimen¬ 
sion lines of the former. Dimensions under 24 
inches are most conveniently given in inches; 
larger dimensions are given in feet and inches. 
The usual practice is to indicate feet and inches 
on drawings by short marks, “prime'' marks ('), 
placed at the right, and a little above the figure, 
one mark (') indicating feet, and two marks, 
“double prime" marks ('), indicating inches, so 
that 5' 7" would read 5 feet 7 inches. Some drafts¬ 
men do not consider this method of marking safe 
enough to eliminate mistakes, and prefer to write 
dimensions of this kind in the form 5 ft. 7". A 
method equally satisfactory in preventing possible 
mistakes is to place a short dash between the 
figure giving the number of feet and that giving 
the number of inches, at the same time retaining 
the “prime" marks; thus, 5'—7". When feet only 
are given, it is well, for the sake of uniformity 
and to prevent any misunderstanding, to give the 
dimension in the form 5'—0". 

A few examples showing the principles of the 
usual methods of dimensioning drawings may be 
of value. In Fig. 73 is shown a simple bushing. 
The diameter of the hole or bore is given as 2 
inches by a dimension line passing through the 
center of the circles in the end view. It is con- 


58 


SELF-TAUGHT MECHANICAL DRAWING 


fusing, however, to have more than one dimension 
line passing through the same center, and, there¬ 
fore, the outside diameters of the bushing have 
been given on the side view. The lengths of the 
various steps or shoulders of the bushing are given 
below the side view, as is also the total length. It 
will be noticed that the dimensions of the three 
steps are slightly offset—that is, the dimension 



Fig. 73.—Simple Example of Dimensioning a Drawing. 


lines do not extend in one straight line; this makes 
a very clear arrangement. 

The method of dimensioning holes drilled in a 
circle is shown in Fig. 74. Outside of the dimen¬ 
sion for the holes themselves only the diameter of 
the circle passing through the centers of the holes 
is given, together with the number of holes. As 
the holes, of course, are to be equally spaced, that 
is all that is required. When a great many bolt 
holes or bolts occur around a flange, it is not nec¬ 
essary to draw them all in on the working draw¬ 
ing; a common method is to show a few, and to 























WORKING DRAWINGS 


59 


draw the circle passing through their centers, the 
pitch circle. The total number of bolts around the 
flange is, of course, also given. A case of this 
kind is illustrated in Fig. 75. When a great many 
holes are drilled in a row, a similar expedient may 



Fig. 74.—Dimensioning Holes Drilled in a Circle. 


be adopted to avoid showing and dimensioning 
all the holes; an illustration of this is shown in 
Fig. 76. 

In Fig. 77 are shown the common methods of 
dimensioning screws and bolts. At A is shown a 
hexagon head bolt, so drawn that three sides of 





















60 SELF-TAUGHT MECHANICAL DRAWING 


the head are visible. Hexagon bolt-heads are 
usually drawn in this manner in all views, irre¬ 
spective of the fact that the rules of projection 
would call for only two sides to be visible in one 
view. The reason for this is partly that the bolt- 


! 



Fig. 75.—Simplified Method of Dimensioning Holes 
Drilled in a Circle. 


head looks better when three sides are visible, and 
partly that when so drawn there can be no confu¬ 
sion whether a hexagon or a square head is meant. 
If only two sides were shown, as at B, the head, 
especially if carelessly drawn, might be mistaken 
for a square bolt-head. As a rule, the dimensions 



















WORKING DRAWINGS 


61 


of bolt-heads are standard for given diameters of 
bolts, and no dimensions are required for the 
head. In some cases, however, the head may be 
required to fit a given size of wrench, or for some 
other reason be required to be made different from 
the standard size; in such cases dimensions may 
be given as shown at C, Fig. 77, the dimension 
“T'hex.^’ indicating that the head is one inch 



Fig. 76.—Dimensioning Holes Drilled in a Row. 


'‘across flats.In the same way, “i" sq.^^ would 
indicate that the head should be square, and three- 
quarters inch “across flats. 

The length of the bolt should be given as shown 
in the lower view in Fig. 77. The dimensions 
should be given “under the head,’^ both the total 
dimension, and the distance to the beginning of 
the thread. 

In general, full circles should be dimensioned by 
their diameters; an arc of a circle, again, should 
be dimensioned by its radius. The center from 
































62 


SELF-TAUGHT MECHANICAL DRAWING 


which the arc is struck should preferably be indi¬ 
cated by a small circle drawn around it. In small 
dimensions, the arrow points are frequently placed 
outside of the lines between which the dimension 
is given, as shown in Fig. 71 in dimensioning 
the narrow ribs; sometimes, the figures giving the 



Fig. 77.— Dimensioning Screws and Bolts. 


dimension are themselves- placed outside of the 
space between the arrow heads, because the space 
is too small to permit the dimension to be clearly 
written within it. 

The principal dimensions should be so given 
that the workman will not have to add a number 
of other dimensions to get them. When the 
dimensioning of a piece naturally divides itself 
into several measurements, an over-all dimension 
should always be given for verification. If, how- 




































































WORKING DRAWINGS 


63 


ever, the piece terminates with a round end, as 
the yoke in Fig. 71, the over-all dimension may 
properly terminate at the center of curvature of 
the end, the distance beyond being of entirely 
secondary importance, and being taken care of by 
its radius. If a dimension has been given in one 
view, there is usually no reason for repeating it 
in the other views; sometimes such repetitions 
would cause too many dimensions to be given in 
each view, so that confusion would arise, and in¬ 
stead of making the drawing plainer, the repeti¬ 
tion of dimensions might cause mistakes which 
otherwise would have been avoided. 

Drawings should always be dimensioned Xh^full 
size of the finished article, regardless of the scale 
to which the drawing is made. If a drawing is 
made to any other scale than full size, it is cus¬ 
tomary to state on the drawing the scale to which 
it is made, as ‘‘Scale, J inch = l ft.’’ 

A drawing should be so marked as to tell the 
workman what surfaces are to be finished; a fin¬ 
ished surface is usually indicated by the letter 
“f” placed either upon the line representing the 
surface, or in close proximity to it. While the 
amount and kind of finish is usually left to the 
workman to determine, the best modern methods 
require that the draftsman should indicate on the 
drawing how closely the various parts are to be 
machined. A very commendable method is to 
give dimensions in thousandths of an inch, where 
accuracy is required, and in common fractions in 
cases where there is no need of working to thou¬ 
sandths. In very highly systematized establish- 


64 SELF-TAUGHT MECHANICAL DRAWING 

ments, the limits of variation between which any 
measurement is allowed to vary, are given with 
each dimension, or, at least, with dimensions for 
diameters which are to fit the holes or bores of 
other pieces. The determination of the limits of 
accuracy required calls for good judgment on the 
part of the draftsman. Limits may be expressed 
in two ways. For instance, a running fit on a 
shaft to go into a IJ inch standard size hole may 
be marked 

-0.0005 max. 

^^-0.0015 min. 

or it may be expressed 

1.4995 max. 

1.4985 min. 

which means that the shaft must not be larger 
than 1.4995 inch, and not smaller than 1.4985 inch. 

On drawings, the tap drill size and the depth of 
tapped holes should always be shown. Surfaces 
to be ground to size should be marked “grind.’’ 
If the surface is to be filed, the words “file finish” 
are substituted for the letter “f.” Finishing 
marks, as a rule, are used on castings and forg¬ 
ings only. On work made from bar stock, every 
surface is nearly always finished, so that here the 
finishing marks are omitted. When a casting or 
forging is finished on every surface, it is not nec¬ 
essary to show finish marks, but the words “finish 
all over” may be written in a conspicuous place, 
so as to readily catch the eye of the workman. If, 
on work made from bar stock, it is desired that 
the piece be left rough at any point, the words 


WORKING DRAWINGS 


65 


‘‘stock size^’ may be applied to the figures giving 
that particular dimension. For instance, on a 
IJ-inch cold rolled shaft, turned for journals for a 
short distance at each end, the central part would 
be dimensioned “IJ-inch stock size.''- 

While the practice of indicating finished surfaces 
by the letter “f" is by far the most frequently 
met with, it is by no means universal. In some 
shops the words “polish," “ream," “finish," etc., 
are written near the lines representing the sur¬ 
faces to be thus treated. Still another method 
much in use is to draw a red line outside of the 
line representing each surface to be finished. If 
a blue-print is made from a tracing thus pre¬ 
pared, the red lines will print fainter than the 
black ones, and the finish lines on the blue-prints 
are traced over with a red pencil or red ink before 
being sent out in the shop. This method, how¬ 
ever, is more expensive than that of indicating 
the finished surfaces by the letter “f," and on 
complicated drawings, the many additional red 
lines tend to cause confusion. By whatever 
method the finish is indicated, the finishing 
marks should always be shown fully in every view 
of the object. 

It frequently happens that the representation of 
an object is made clearer by the use of sectional 
views, representing the object as having been cut 
in two, either wholly or in part. Examples of this 
are shown in Figs. 69, 70 and 71. From these 
illustrations it is apparent that the construction of 
the various pieces is much more clearly exhibited 
when a section is shown. The surface “cut" or 




66 SELF-TAUGHT MECHANICAL DRAWING 

shown in section is cross-hatched or cross-sectioned 
with fine lines at a distance apart varying from a 
thirty-second to an eighth of an inch, according 
to the size of the drawing and the piece. The 
cross-sectioning brings the parts in section into 
bold contrast with the remainder of the drawing, 
and prevent all confusion as to what parts are in 
section and what parts shown in full. All lines 
beyond the sectional surface which are exposed to 
view, should be shown in the drawing as usual. 
Should it be deemed necessary, which it seldom is, 
to show any parts that have been cut away for the 
purpose of showing a section, such parts may be 
drawn in by dash-dotted lines, this indicating that 
the parts thus shown are in front of the section 
and actually cut away. 

When a mechanism is shown in section, the dif¬ 
ferent parts of the same pieces should always be 
cross-sectioned by lines inclined in the same direc¬ 
tion, while separate pieces adjoining each other 
should always, when possible, be cross-sectioned 
by lines running in different directions. When a 
solid round piece is exposed to view by a section, 
it is customary to show it solid, and not to section 
it; the screw stud in Fig. 69 is an example of this 
practice. 

Sectional views may also be used for many pur¬ 
poses where a slight deviation from the theory of 
projection will tend to simplify the representation 
of certain machine details. The shape of the arm 
of a pulley or gear, or of any other part of a cast¬ 
ing, may be conveniently represented in this way. 
The cutting plane may be assumed to lie at any 


WORKING DRAWINGS 


67 


angle necessary to bring out the details most clear¬ 
ly. A sectional view, for instance, may represent 
a casting as though it were cut through partly on 
one plane and partly on another. In all such cases, 
however, it should be indicated in another view of 
the object‘just where the sectional views are sup- 



Fig. 78.—Methods of Showing Sections. 


posed to be taken, so that no confusion may arise 
on this account. The examples in the following 
will serve to make clear the principles laid down. 

In Fig. 78 are shown sections of two hand- 
wheels. When an object is symmetrical it is 
unnecessary to show more than one half in sec¬ 
tion, although it is quite common to section gears, 
pulleys, etc., completely on working drawings. 
The hand-wheel at A in Fig. 78 is represented as 



















































68 SELF-TAUGHT MECHANICAL DRAWING 


though cut in two along its diameter BC, When 
the section is taken along the center line, it is not 
absolutely necessary to explain where the section 
is taken; but it can do no harm to make a practice 
of in all cases to state where the section is made, 
except when perfectly obvious. In this case it 
would be clear that the section is taken through 


SECTION AT A*B 

Fig. 79.—A Gear-wheel in Section. 

the center, and the legend “Section at is 

given only to show the principle. The hand-wheel 
at D is provided with four arms, and the method 
of representing the shape of the arms, hub and 
rim are clearly indicated. 

In Fig. 79 are shown two views of a gear-wheel, 
indicating the conventional method of represent¬ 
ing gears on drawings. The view on the left side 
is the side view, and as all the teeth are, of course. 

































WORKING DRAWINGS 


69 


alike, it is unnecessary to draw more than a few 
of them. The pitch line of the teeth is represented 
by a dash-dotted line. In the part of the gear¬ 
wheel rim where the teeth are not shown, the face 
of the gear is indicated by a solid line, and the 
bottom of the teeth by a dotted line. In the case 
of machine-cut gearing, where the teeth are cut 
by'standard formed cutters, it is unnecessary to 
show any teeth at all on the rim of the gear, it 
being sufficient to state the pitch and the number 
of teeth, as will be more fully explained later in 
the chapter on gearing. To show the shape to 
which the arms are formed, a sectional view of 
one of the arms is drawn in the side view; the 
ends of the shaft are supposed to be broken off, 
and are, therefore, sectioned as shown. The right- 
hand view of the gear is a section taken along the 
line AB. It will be noted that the shaft and key 
are not sectioned, usual practice being followed in 
this respect. The gear shown has five arms, and 
the line AB cuts through one of them only. This 
arm, however, is not sectioned in the right-hand 
view, and two opposite arms are drawn as though 
both of them lay in the plane of the paper. While 
this is not theoretically correct, it is the method 
usually followed because of simplicity in drawing 
and clearness of representation. The method of 
representing the gear teeth in the sectional view 
is the one commonly employed. 

Sectional and top views of a cylinder end with 
flange and cover are shown in Fig. 80. This 
cylinder cover has only five bolts, and the plane 
through which the section is taken cuts through 


70 


SELF-TAUGHT MECHANICAL DRAWING 


only one of the bolts. It is common practice, how¬ 
ever, to draw the section as shown at the left. 
The bolts are shown as if two of them were in the 
plane of the section. The bolts are not sectioned, 


j 





Fig. 80.—Section of Cylinder End with F’ange and Cover. 


but are drawn in full, as explained previously. 
Dotted lines of the remaining bolts, or full lines 
of their nuts, should not be shown, because this 
detracts from the clearness of the drawing; the 
top view shows clearly the number of the bolts 
and their arrangement, and that is all that is nec¬ 
essary. Some draftsmen prefer to draw sections 
























































WORKING DRAWINGS 


71 


of this kind as indicated at the right in Fig. 80. 
This method, however, is not as commonly used. 

In a case where the object is rather unsymmet- 
rical, as, for instance, in Fig. 81, the draftsman’s 
judgment must often be relied upon to decide how 



Fig. 81. — Another Method of Showing Sections. 

it shall best be shown in section. Usually the 
sectional view is made symmetrical as shown, the 
distances A in the lower view being made equal to 
the radius A in the top view. 

The materials for the various details making up 
a complete mechanism are usually cross-sectioned 





























72 SELF-TAUGHT MECHANICAL DRAWING 


in such a way as to indicate the material from 
which each piece is made. There is, however, no 
universally adopted or recognized standard for 
cross-sectioning for the purpose of indicating dif¬ 
ferent materials. In Fig. 82 is shown a chart, 



WROUGHT IRON 



STEEL 



MALLEABLE IRON 




BRONZE 


BRASS 


COPPER 


LEAD 





Fig. 82.—Cross-sectioning used for Indicating Different 

Materials. 


published by Mr. I. G. Bayley in Machinery, Oc¬ 
tober, 1906, which represents average practice, 
although it must be distinctly understood that 
there is no agreement in all respects between the 
numerable charts in use in various drafting-rooms. 
For this reason, cross-sectioning alone should 
never be depended upon for indicating to the work- 























WORKING DRAWINGS 


73 


man the kind of material to be used. Written 
directions should also be given, the kind of mate¬ 
rial for each part being plainly marked. Tool steel 
may be abbreviated “T. S.”, machine steel, “M. 


1 

- 


f ■■ ■ 


ROUND BAR, SOLID 



ROUND BAR, HOLLOW 


SQUARE OR RECTANGULAR BAR 



I-BEAM 


Fig. 83.—“Broken’' Drawings of Long Objects. 

• 

wrought iron, “W. I.^’; cast iron, '‘C. I.^', 
etc. The less common materials in machine con¬ 
struction, such as bronze, brass, copper, etc., should 
preferably be written out in full, in order to avoid 
any chances for confusion. It is better to be too 


































































74 SELF-TAUGHT MECHANICAL DRAWING 

explicit as regards the information on the draw¬ 
ing, than to risk misunderstandings and conse¬ 
quent errors. 

Long bars, shafting, structural beams, etc., can¬ 
not conveniently be shown for their full length on 
the drawing. In such cases the pieces are drawn 
as long as the drawing and the adopted scale per¬ 
mit, and are broken as shown in Fig. 83, a part 
between the two end portions shown being imag¬ 
ined as broken out. The di¬ 
mensions, of course, are given 
for the full length of the piece, 
as if not broken. 

There are several conven¬ 
tional methods for showing 
screw threads; these methods 
are adopted largely for saving 
of time, as it would be out of 
the question to spend the time 
required for drawing a true 
helical screw thread on a work¬ 
ing drawing. A method for 
very nearly approximating the 
appearance of a theoretically correct screw drawing 
is shown in Fig. 84, where the projection of the 
screw helix is drawn by straight lines. The V- 
shaped outline is first laid out, and the connecting 
lines are then drawn. It will be noticed that the 
lines representing the roots of the threads are not 
parallel with those representing the tops or points. 
This aids in making the‘drawing resemble that of 
a true helix. 

Usually, however, much simpler methods are 



Fig. 84.—Method of 
Drawing a Screw, 
Giving Correct He¬ 
lix Effect. 






WORKING DRAWINGS 


75 


employed for indicating screw threads. In Fig. 
85, A, ^ and C, some of these methods are shown. 
When a long piece is threaded the entire length, 
this fact can be indicated as at D, which saves 
drawing the conventional thread for the full length 
of the piece. The lines indicating the thread are 



E F 

Fig. 85.—Simplified Methods for Showing Screw Threads. ' 


inclined, the same as would be the lines represent¬ 
ing the true helix. At E in Fig. 85 is shown a 
right-hand thread and at F' a left-hand thread, the 
different direction of inclination of the thread in¬ 
dicating this fact. However, if a thread is to be 
left-hand, it should always be so marked on the 
drawing. It is usual to abbreviate left-hand, writ¬ 
ing‘‘L. 
















































































































































76 SELF-TAUGHT MECHANICAL DRAWING 

Three methods of indicating tapped holes are 
shown in Fig. 86, these being used when the holes 
are obscured from view, and shown by dotted lines. 
When a tapped hole is shown in section, and looked 
upon from the top, it is shown as indicated at D, 
while if seen from the side, in section, it is repre¬ 




sented as at E. A surface having tapped holes in 
it, seen from above, is shown at F. At G and H 
are shown the methods of representing bolts or 
screws inserted in place in tapped holes. It will 
be noted that when the threads of a tapped hole 
are exposed to view by section, the lines repre¬ 
senting the screw helix will be seen to slope in the 
opposite direction to those of the screw, it being 




















































WORKING DRAWINGS 


77 


the back side that is exposed to view. An example 
of this is shown in Fig. 71 as well as in Fig. 86. 

In drawings made for use in the shop it is cus¬ 
tomary to make the lines of uniform thickness. 
For shop use such drawings are as good as any. 
When, however, the purpose of a drawing is chiefly 
to show up the object which it represents, its ef¬ 
fectiveness may be considerably enhanced by the 
use of shade lines as shown in Fig. 87. In shade 
line work, the light is usually assumed to come 
from the upper left hand corner, 
and to shine diagonally across 
the paper at an angle of forty- 
five degrees. Lines on the side 
of the object away from the 
light, or lines separating light 
from dark surfaces, are made 
extra heavy. This gives to the 
drawing a suggestion of relief. 

An examination of the lines of 
Fig. 87 taken in connection with the direction from 
which the light is supposed to come will show, 
without the aid of any other view, that the hex¬ 
agonal part is raised above the surface of the 
square, and that the circle in the center represents 
a depression. 

When a drawing is intended for permanent use 
it is customary to make only a pencil layout on 
paper, usually on brown paper, and from this to 
make a tracing from which any number of blue 
print copies may be made. The tracing is usually 
made on the regular tracing cloth. This has one 
glazed and one unglazed surface. Either surface 



Fig. 87.—Use of 
Shade Lines. 







78 SELF-TAUGHT MECHANICAL DRAWING 

may be used. The tracing cloth is drawn tightly 
over the pencil drawing, and its surface is cleaned 
of any greasiness with dry powdered chalk. This 
insures a good flow to the ink. In doing the ink 
work curved lines should be made first, straight 
lines afterwards, as mentioned in Chapter I. 

The blue prints are made in the same manner as 
photographs are printed, the tracing taking the 
place of the photographic negative. An exposure 
of from three to ten minutes may be required, de¬ 
pending on the freshness of the blue print paper 
and the brightness of the sun. After the proper 
exposure has been given, which may require some 
experimenting at first, until one gets accustomed 
to the change in the paper which the light makes, 
the print is thoroughly rinsed out in clear water 
and dried, by being hung up by one edge. 

White writing may be made on a blue print with 
saleratus water, the water being given all the sale- 
ratus it will dissolve. 


CHAPTER VI 


✓ 


ALGEBRAIC FORMULAS 

In order to be able to carry out the calculations 
required in simple machine design, it is necessary 
that a general understanding of the use of for¬ 
mulas, such as are used in mechanical hand-books 
and in articles in the technical press, is acquired. 
Knowledge of algebra or so-called “higher mathe¬ 
matics’^ is by no means necessary, although, of 
course, such knowledge is very valuable; but simple 
formulas can be used, and the results of scientific 
results employed in practical work to a very great 
extent, by any man who understands how to use 
the formulas given by the various autnorities; and 
the knowledge required for an intelligent use of 
algebraic formulas can be very easily acquired. 
All the mathematical knowledge necessary as a 
foundation is a clear understanding of the funda¬ 
mental rules and processes of arithmetic. 

A formula is simply a rule expressed in the sim¬ 
plest and most compact manner possible. By using 
letters and signs in the formula instead of the 
words in the rule, it is possible to condense, in a 
very small space, the essentials of long and cum¬ 
bersome rules. The letters used in formulas sim¬ 
ply stand in place of the figures which would be 
used for solving any specific problem; the signs 
used are the ordinary arithmetical signs used in 

79 


80 SELF-TAUGHT MECHANICAL DRAWING 

all kinds of calculations. As each letter stands for 
a certain number or quantity, whenever a specific 
problem is solved the figures for that case are put 
into the formula in place of the letters, and the 
calculation is carried out as in ordinary arithmetic. 
This may, perhaps, be made clearer by means of a 
few examples. 

The circumference of a circle equals the diameter 
times 3.1416. This rule may be written as a 
formula as follows: 

C= D X 3.1416. 

In this formula C = circumference, and D = 
diameter. No matter what the diameter is, this 
formula says, the circumference is always equal to 
the diameter {D) times 3.1416. Assume that the 
diameter is 5 inches. Then, to find the circumfer¬ 
ence, place 5 in the formula in place of D. 

C = 5 X 3.1416 = 15.708 inches. 

If the diameter of a circle is 12 feet, then 
C = 12 X 3.1416 = 37.6992 feet. 

This, of course, is the very simplest kind of a 
formula, but it illustrates the principle involved, 
and indicates how easily formulas may be em¬ 
ployed. 

One of the most well-known formulas in steam 
engineering is that giving the horse-power of an 
engine, when the average or mean effective pres¬ 
sure of the steam on the piston, the length of the 
stroke of the piston in feet, the area of the piston 
in square inches, and the number of strokes per 
minute, are known. Let 


ALGEBRAIC FORMULAS 


81 


H.P. = horse-power, 

P = mean effective pressure in pounds per 
square inch, 

L = length of stroke in feet, 

A = area of piston in square inches, and 
N = number of strokes per minute. 


Then 


KP, = 


PX LX AX N 

33,000 


The rule conveying this information expressed 
in words would require considerable space, and be 
difficult to grasp immediately; but the meaning of 
the formula is quickly understood. If the pressure 
(P) equals 75 pounds, the stroke (L) 2 feet, the 
area of the piston (A) 125 square inches, and the 
number of strokes per minute {N) 60, then 


H.P. = 


75 X 2 X 125 X 60 

33,000 


= 34.1. 


It will be seen that the values for the different 
quantities are merely inserted in the formula in 
place of the corresponding letters, and then the 
calculation is carried out as usual. It will be 
remembered that the line between numerator and 
denominator in a fraction also means a division; 
that is 

= 1,125,000 ^ 33,000 = 34.1. 


It is very common in formulas to leave out, en¬ 
tirely, the sign of multiplication (X) between the 
letters expressing the values of the various quanti¬ 
ties that are to be multiplied. Thus, for example. 





82 SELF-TAUGHT MECHANICAL DRAWING 


PL means simply P X L, and if P = 21 and L == 3, 
then PL = P X L = 21 X3 = 63. If the multipli¬ 
cation signs are left out in the formula for the 
horse-power of engines just referred to, the for¬ 
mula 


PX L X A>^N 
33,000 


could be written 


PLAN 

33,000* 


As a further example of the leaving out of the 
multiplication sign in a formula, assume that D 
= 12, P = 3, and r = 2, then 

DRr ^ D'xRXt _ 12 X 3 X 2 _ 72 _ ^ 

9 9 9 9 


It must be remembered that no other signs, ex¬ 
cept the multiplication sign, may thus be left out 
between the letters in a formula. 

From the examples given, the use of simple 
formulas is clear; each letter stands for a cer¬ 
tain number or quantity which must be known in 
order to solve the problem; when the formula is 
used for the solution of a problem, the letters are 
simply replaced by the corresponding number, 
and the result is found by regular arithmetical 
operations. 

The expressions “square^^ and “square rooF^ 
and “cube"’ and “cube roof are frequently used 
in engineering hand-books and technical journals. 
It would seem, to one unfamiliar with these names 
and their mathematical meaning, as well as the 
signs by which they are indicated, that difficult 
mathematical operations are involved; but this is 
not necessarily always the case. The square of a 
number is simply the product of that number mul- 








ALGEBRAIC FORMULAS 


83 


tiplied by itself. Thus the square of 3 is 3 X 3 = 9, 
and the square of 5 is 5 X 5 = 25. In the same 
way, the square of 81 is 81 X 81 = 6561. Instead 
of writing 81 X 81, it is common practice in 
mathematics to write 81'^ which is read '‘81 
square, and indicates that 81 is to be multiplied 
by itself. Similarly, we may write = 7 X 7 = 
49, and 12' = 12 X 12 = 144. The little "2’Mn the 
upper right-hand corner of these expressions is 
called "exponent. Nearly all mechanical and 
engineering hand-books are provided with tables 
which give the squares (and also the square root, 
cube and cube root) of all numbers up to 1000, so 
that it is usually unnecessary to calculate these 
values by actual multiplication. 

As the squares of numbers are frequently used 
in formulas for solving problems occurring in 
machine design and machine-shop calculations, a 
few examples will be given below of formulas con¬ 
taining squares. 

The area of a circle equals the square of the 
radius multiplied by 3.1416. Expressed as a for¬ 
mula, if A = area of circle, R = radius, and the 
Greek letter n (Pi) = 3.1416, we have: 

A = t:. 

If we want to know the area of a circle having a 
5-foot radius, we -have: 

A = 5“7:=5X5X 3.1416 = 78.54 square feet. 

As a further example, assume a formula to be 
given as follows: 

^ = i)ir^ 


1 



84 SELF-TAUGHT MECHANICAL DRAWING 


Assume that D = 3, AT = 5, i? = 4, and n (as 
usual) =3.1416. What is the value of A? Insert¬ 
ing the values of the various letters in the formula, 
we have: 

’ A = 3" X 5 + 4^ X TT ^ 3X3X5 + 4X4X71 ^ 
3X4 3X4 

9 X 5 + 16 X TT _ 45 + 50.2656 _ 95.2656 _ ^ 

12 12 12 A 9388. 

It will be seen in the example above that all the 
multiplications are carried out before any addition 
is made. This is in accordance with the rules of 
mathematics. When several numbers or expres¬ 
sions are connected with signs indicating that 
additions, subtractions, multiplications or divisions 
are to be made, the multiplications should be 
carried out before any of the other operations, 
because the numbers that are connected by the 
multiplication sign are actually only factors of 
the product thus indicated, and consequently this 
product must be considered as one number by 
itself. The other operations are carried out in the 
order written, except that divisions when written 
in line with additions and subtractions, precede 
these operations. A number of examples of these 
rules are given below: 

12 X 3 + 7 X 2i - li= 36 + 174 - li = 52. 

5 + 13 X 7 - 2= 5 + 91 - 2 = 94. 

9 - 3 + 9 X 3= 3 + 27 = 30. 

9 + 9- 3- 2= 9 + 3- 2 = 10. 

Sometimes, however, in formulas, it is desired 
that certain operations in addition and subtraction 








ALGEBRAIC FORMULAS 


85 


precede the multiplications. In such cases use are 
made of the parenthesis ( ) and bracket [ ]. These 
mathematical auxiliaries indicate that the expres¬ 
sion inside of the parenthesis or bracket should be 
considered as one single expression or value, and 
that, therefore, the calculation inside the parenthe¬ 
sis or bracket should be carried out by itself com¬ 
plete before the remaining calculations are com¬ 
menced. If one bracket is placed inside of another, 
the one inside is first calculated, and when com¬ 
pleted the other one is carried out. Some examples 
will illustrate these rules and principles: 

(6-2) X 3 + 4 =4 X 3 + 4 = 12+ 4 = 16. 

3 X (12 + 7) - 28i = 3 X 19 - 28i = 57-28i =2. 

3 + [5 X 3 (5 + 2) - 3] X 6 = 3 + [5 X 3 X 7 -3] 
X 6 = 3 + [105 - 3] X 6 = 3 + 102 X 6 = 3 + 612 
= 615. 

Without the parentheses and brackets, the calcu¬ 
lations above would have been as follows: 

6-2X3 + 4 = 6- 6 + 4 = 4. 

3 X 12 + 7 - 28i = 36 + 0.2456 = 36.2456. 

3 + 5X3X5 + 2- 3X6 = 3 + 75 + 2- 18 = 62. 

These examples should be carefully studied until 
thoroughly understood. 

We are now ready to return to the question of 
square roots. The square root of a number is that 
number which, if multipled by itself, would give 
the given number. Thus, the square root of 9 is 3, 
because 3 multiplied by itself equals 9. The square 
root of 16 equals 4, of 36 equals 6, and so forth. It 
will be seen at once that the square root may be 


86 SELF-TAUGHT MECHANICAL DRAWING 

considered, or, rather, actually is the reverse of 
the square, so that if the square of 20 is 400, then 
the square root of 400 is 20. In the same way, as 
the square of 100 is 10,000, so the square root of 
10,000 is 100. The sign used_in mathematical 
formulas for the square root is V . Thus V 9 = 3, 
V 49 = 7, and so forth. The process of actually 
calculating the square root is rather cumbersome, 
and it is very seldom required, because, as already 
mentioned, the engineering hand-books usually 
give tables of square roots for all numbers up to 
1000, and for larger numbers the tables can also be 
used for obtaining the square root approximately 
correct, or at least near enough so for almost all 
practical calculations. 

The cube of a number is the product resulting 
from repeating the given number as a factor three 
times. Thus, the cube of 3 is 3 X 3 X 3 = 27, and 
the cube of 17 is 17 X 17 X 17 = 4913. In the same 
way as we write 2^ = 2 X 2 = 4, for the square of 
2, so we can write 2^ = 2X2X2 = 8, for the cube 
of 2. The exponent (^) indicates how many times 
the given number is to be repeated as a factor. 
The cube of 4, for example, may be written 4^ = 4 
X 4 X 4 = 64. Similarly 17^ = 4913. The expres¬ 
sion 17^ may be read “the cube of 17,’^ “17 cube, 
or “the third power of 17.^’ In the same way as 
the square root means the reverse of square, so the 
cube root (or “third root”) means the reverse of 
cube or “third power”; that is, the cube root of a 
number is the number which, if repeated as factor 
three times, would give the given number. For 
example, the cube root of 64 is 4, because 4 X 4 X 


ALGEBRAIC FORMULAS 


87 


4 = 64. It is evident that if the cube of a number, 
say 6, is 216 (6X6X6 = 216), then the cube root 
of 216 is 6. The sign used in formulas for the cube 

root is f . For^ample, 8=2 (because 2X2 
X 2 = 8), and 125 = 5 (because 5 X 5 X 5= 125). 
Similarly, 1^3,723,875 = 155. 

The use of the square and square root, and cube 
and cube root in formulas may be shown by a few 
examples: _ 

A = X i^ C 
C'^ + D'^ * 

Assume that 5 = 27, C = 25, and D = 2. Insert 
these values in the formula. Then 

_ X ^ 3X5 ^ _15 
25'^+ 2* 125 + 4 129 

As another example: y' 

^ + C^ + Z)^ 

B^XiC 


Assume B = 2, C = 9, and D = L Then 

. _ 2^~l~9^ 4'4 ^ _ 4 ~1~ 81 ~l~ 16 _ 101 

~ 2^Xl/9~ ~ 8X3 “ 24 ‘ 


4.208. 


In the same way as2^ = 2X2 = 4, so2^ = 2X2 
X 2 X 2 = 16, and 2*" = 2 X 2 X 2 X 2 X 2 = 32. 

The expression 2^ is read the '‘fourth power of 
2,^^ and 2^ the “fifth power of 2.’’ The exponents 
(^) and (*') indicate how many times the given 
number is to be repeated as factor. 

If, again, it is required to find the number which, 
if repeated as factor four times, gives the given 

number, we must obtain the “fourth root’^ or V . 















88 SELF-TAUGHT MECHANICAL DRAWING 


Thus, V 16 = 2, because 2 X 2 X 2 X 2 = 16. In 
the same way V 256 = 4. The fifth root is writ¬ 
ten V ; and V 243 = 3, because 3X3X3X3X3 
= 243. 

These explanations, when fully understood, will 
eliminate all difficulties with formulas of a simple 
nature, and with such expressions as cube root, 
exponents, etc. 

An important method facilitating the use of 
formulas, is commonly known as the transposition 
of formulas. A formula for finding the horse¬ 
power which can safely be transmitted by a gear 
of a given size, running at a given speed, is: 

TT P = DXNXPXFX2m 

• * 126,050 

In this formula H.P. = horse-power, 

D = pitch diameter, 

N = revolutions per minute, 

P = circular pitch of gear, 

F = width of face of gear. 


Assume, for example, that the pitch diameter of 
a gear is 31.5 inches, the number of revolutions 
per minute 200, the circular pitch IJ inch, and the 
width of the face 3 inches. Then, if these values 
are inserted in the formula, we have: 


H.P. = 


31,5 X 200 X Ij X 3 X 200 
126,050 


= 45 horse¬ 


power, very nearly. 

Assume, however, that the horse-power required 
to be transmitted is known, and that the pitch of 
the gear is required to be found. Assume that 


( 






ALGEBRAIC FORMULAS 


89 


H.P = SO; D = 31.5; N = 200 ; F = S; and that P 
is the unknown quantity; then, inserting the 
known values in the formula, gives us: 

QO = 3 1.5 X 200 X P X 3 X 200 

126,050 


In order to be able to find P, we want it given 
on one side of the equals sign, with all the known 
quantities on the other side. If we multiply the 
expressions on both sides of the equals sign by 
the same number we do not change the conditions; 
thus 


30 X 126,050 = 


31.5 X 200 X P X 3 X 200 X 126,050 

126,050 


By canceling the number 126,050 on the right- 
hand side we have: 


30 X 126,050 = 31.5 X 200 X P X 3 X 200. 

If we now divide on both sides of the equals 
sign with 31.5 X 200 X 3 X 200, we have: 

30 >? 1 26,0 50 _ 31.5 X 2 00 X P X 3 X 200 

31.5 X 200 X 3 X 200 31.5 X 200 X 3 X 200 * 


We can now cancel all numerical values in the 
fraction on the right-hand side; then: 

_^30_X 126,050__ p 

31.5 X 200 X 3 X 200 


This is then the transposed formula giving 
and from this we find that P = 1 inch. 

In general, any formula of the form 



B 

C' 



can be transposed as below: 

AXC=B; C = “. 












90 


SELF-TAUGHT MECHANICAL DRAWING 


It will be seen that the quantities which are in 
the denominator on one side of the equals sign, are 
transposed into the numerator on the other side, 
and vice versa. 


Examples: 


A 


Then: 

n = 


BXC 

A 


B 


BXC 

D • 

AXD 
C ’ 



AXD 

B 


Then: 


A = 


EXFXG 
KXL • 


_ AXKXL 
EX G 



AX KXL 
EXG 



AX K XL _ 

EXE ’ 


K = 


EXFXG 

AXL 



EXFXG 

AXK 


The principles of transposition of formulas can 
best be grasped by a careful study of the examples 
given. Note that the method is only directly ap¬ 
plicable when all the quantities in the numerator 
and denominator are factors of a product. If con¬ 
nected by + or — signs, the transposition cannot 
be made by the simple methods shown unless the 
whole sum or difference is transposed. Example: 

A = thenZ) andB + C == A X D. 

The most usual caclulations, perhaps, in some 
classes of machine design, are those involving the 
finding of the strength of certain machine mem¬ 
bers; and, in order to find the strength of these 














ALGEBRAIC FORMULAS 


91 


members, it is necessary to first find the cross- 
sectional area of the part subjected to stress. For 
this reason, the remainder of this chapter will be 
largely taken up with rules and formulas for find¬ 
ing the areas and other properties of various geo¬ 
metrical figures. Rules and formulas for volumes 
of solids will also be given. Examples have been 
given in some cases merely to show the applica¬ 
tions of the formulas. 

The area of a triangle equals one-half the prod¬ 
uct of its base and its altitude. The base may be 
any side of the triangle, and the altitude is the 
length of the line drawn from the angle opposite 
the base, perpendicular to it. 

Assume that A = area of triangle, 

B = base, 

H = altitude. 


Then the rule above may be expressed as a 
formula as follows: 

BXR 


Let the base (B) of a triangle be 5 feet, and the 
altitude (H) 8 feet. Then the area 


A 


2 


40 


= 20 square feet. 


The area of a square equals the square of its 
side. If A = the area, and S the side of the square, 
then 

A = 


If the side is 9.7 inches long, then 
^ = 9,7-'=9.7 X 9.7 =94.09 square inches. 



92 SELF-TAUGHT MECHANICAL DRAWING 


The area of a rectangle equals the product of its 
long and short sides. If ^ = area, L = length of 
the longer side, and i7= length of the shorter side, 
then 

A = LX H. 


The area of a parallelogram equals the product 
of the base and the altitude. 

The area of a trapezoid equals one-half the sum 
of the parallel sides multiplied by the altitude. 
If A = area, B = length of one of the parallel sides, 
C = length of the other parallel side, and H = 
altitude, then 



B + C 

2 


X H. 


Assume that the lengths of the two parallel 
sides are 12 and 9 feet, respectively, and that the 
altitude is 16 feet. Then 

A = 2 ^ 16 = 10.5 X 16 = 168 square feet. 

To find the area of an irregular figure bounded 
by straight lines, divide the figure into triangles, 
and find the area of each triangle separately. The 
sum of the areas of all the triangles equals the 
area of the figure. ^ 

The circumference of a circle equals its diameter 
multiplied by 3.1416. 

The diameter of a circle equals the circumfer¬ 
ence divided by 3.1416. 

The area of a circle equals the square of the 
diameter multiplied by 0.7854. 

The diameter of a circle equals the area divided 




ALGEBRAIC FORMULAS 93 


by 0.7854, and the square root extracted of the 
quotient. 

If Z) = diameter, C = circumference, and A = 
area, these last rules may be expressed in formulas 
as follows: 


C = Z)X 3.1416. 


A = D'^X 0.7854. 


D = 
D = 


C 

3.1416* 

IZH 

\ 0.7854 


The length of a circular arc equals the circum¬ 
ference of the circle, multiplied by the number of 
degrees in the arc, divided by 360. If L = length 
of arc, C = circumference of circle, and N = num¬ 
ber of degrees in the arc, then 


CXN 

360 • 


The area of a circular sector equals the area of 
the whole circle multiplied by the quotient of the 
number of degrees in the arc of the sector divided 
by 360. If a = area of sector, A = area of circle, 
and N = number of degrees in sector, then 


a = A X 


JL 

360* 


The area of a circular segment equals the area of 
the circular sector formed by drawing radii from 
the center of the circle to the extremities of the 
arc of the segment, minus the area of the triangle 
formed by these radii and the chord of the arc of 
the segment. 

The area of a pentagon (regular polygon having 






94 SELF-TAUGHT MECHANICAL DRAWING 

five sides) equals the square of the side times 
1.720. 

The area of a hexagon (regular polygon having 
six sides) equals the square of the side times 
2.598. 

The area of a heptagon (regular polygon having 
seven sides) equals the square of the side times 
3.634. 

The area of an octagon (regular polygon having 
eight sides) equals the square of the side times 
4.828. 

The volume of a cube equals the cube of the 
length of its side. 

The volume of a prism equals the area of the 
base multiplied by the altitude. 

The volume of a cylinder equals the area of its 
base circle multiplied by the altitude. 

The volume of a pyramid or cone equals the area 
of the base times one-third the altitude. 

The area of the surface of a sphere equals the 
square of the diameter multiplied by 3.1416. 

The volume of a sphere equals the cube of the 
diameter times 0.5236. 

The volume of a spherical sector equals two- 
thirds of the square of the radius of the sphere 
multiplied by the height of the contained spherical 
segment, multiplied by 3.1416. li V = volume of 
sector, i? = radius of sphere, and iJ= height of the 
contained spherical segment, then 

V ^ YR‘‘ X H X 3.141G. 

Assume that the length of the radius of a spheri- 


ALGEBRAIC FORMULAS 95 


cal sector is 6 inches, and the height of the con¬ 
tained segment 2 inches. Then 

V = ^-X 6'^ X 2 X 3.1416 = 150.7968 cubic inches. 

o • 

The volume of a spherical segment equals the 
radius of the sphere less one-third the height of 
the segment, multiplied by the square of the 
height of the segment, multiplied by 3.1416. If B 
= radius, H = height, and V = volume of segment, 
then 

V = X X 3.1416. 


Assume that the length of the radius is 4 inches, 
and the height of the segment 3 inches. Then 

F = (4- 


^ X 3''' X 3.1416 = 84.8232cubic inches. 


The area of an ellipse equals the long axis multi¬ 
plied by the short axis, multiplied by 0.7854. If 
the area =A, the long axis and the short axis 
= C, then 

A = BXCX 0.7854. 


If the long axis is 12 inches and the short axis 
8J inches, then 

A = 12 X 8J X 0.7854 = 78.54. 

Formulas and application of formulas have not 
been given for such rules which are so simple and 
easy to understand that the reader without diffi¬ 
culty can formulate his own formula. 



CHAPTER Vir 


ELEMENTS OF TRIGONOMETRY 

Trigonometry is a very important part of the 
science of mathematics, and deals with the deter¬ 
mination of angles and the solution of triangles. 
In order to fully understand the subjects treated 
of in the following, it is necessary that the reader 
is fully familiar with the usual methods of desig¬ 
nating the measurements or sizes of angles. While 
mathematicians employ also another method, in 
mechanics angles are measured in degrees and 
subdivisions of a degree, called minutes. The 
minute is again subdivided into seconds, but these 
latter subdivisions are so small as to permit of 
being disregarded in general practical machine 
design. 

A degree is 1-360 part of a circle, or, in other 
words, if the circumference of a circle is divided 
into 360 parts, then each part is called one degree. 

If two lines are drawn from the center of the 
circle to the ends of the small circular arc which 
is 1-360 part of the circumference, then the angle 
between these two lines is a 1-degree angle. A 
quarter of a circle or a 90-degree angle is called a 
right angle. The meaning of obtuse and acute 
angles has already been explained in Chapter II. 
Any angle which is not a right angle is called an 
oblique angle. 


96 


ELEMENTS GF TRIGONOMETRY 97 

A minute is 1-60 part of a degree, and a second 
1-60 part of a minute. In other words, one circle 
= 360 degrees, one degree = 60 minutes, and one 
minute = 60 seconds. The sign (°) is used for in¬ 
dicating degrees; the sign (') indicates minutes, 
and the sign (") seconds. A common abbreviation 
for degree is'‘deg.’’; for minute, "min.”; and for 
second, "sec.” 

Two angles are equal when the number of de¬ 
grees they contain is the same. If two angles are 
both 30 degrees, they are equal, no matter how 
long the sides of the one may be in relation to the 
other. 

Of all triangles, the right-angled triangle occurs 
most frequently in machine design. A right-ang¬ 
led triangle is one having the angle between two 
sides a right angle; the angles between the other 
sides may be of any size. In the calculations in¬ 
volved in solving right-angled triangles, a useful 
application of the squares and square roots of 
numbers is also presented. Assume that the lengths 
of the sides of a right-angled triangle, as shown 
in Fig. 88 , are 5 inches, 4 inches, and 3 inches, 
respectively. Then 

52 ^ 424 . 32 ^ or 25 = 16 + 9 . 

This relationship between the three sides in a 
right-angled triangle holds good for all right-ang¬ 
led triangles. The square of the side opposite the 
right angle equals the sum of the squares of the 
sides including the right angle. Assume, for ex¬ 
ample, that the lengths of the two sides including 
the right angle in a right-angled triangle are 12 


98 SELF-TAUGHT MECHANICAL DRAWING 

and 9 inches long, respectively, as shown in Fig. 
89, and that the side opposite the right angle, the 
hypotenuse, is to be found. We then first square 
the two given sides, and from our rule, just given, 
we have that the sum of the squares equals the 
square of the side to be found. The square root 



of the sum must then equal the side itself. Carry¬ 
ing out this calculation we have: 

12^9' = 144 + 81 = 225 
V 225 = 15 inches = length of hypotenuse. 

Similar methods may be employed for finding 
any of the sides in a right-angled triangle if two 
sides are given. If the hypotenuse were known 
to be 15 inches, and one of the sides including the 
right angle 9 inches, as shown at D in Fig. 90, 
then the other side including the right angle can 
be found. In this case, however, we must subtract 
the square of the known side including the right 















ELEMENTS OF TRIGONOMETRY 99 

angle from the square of the hypotenuse to obtain 
the square of the remaining including side. We, 
therefore, have: 

152^92 = 225-81 = 144 

Vl44 = 12 inches = length of unknown side. 

In the same way, if the lengths 15 and 12 were 



known, we could find the side ACy as shown at Ey 
Fig. 90: 

15^ - 12 - = 225 - 144 = 81 
V81 = 9 inches = length of AC. 

From these examples we may formulate rules 
and general formulas for the solution of right- 
angled triangles when two sides are known. In 
Fig. 91, at F, the square of AB plus the square of 
AC equals the square of BC; the square of BC 
minus the square of equals the square of AB; 
and the square of BC minus the square of AB 












100 SELF-TAUGHT MECHANICAL DRAWING 


equals the square of AC. These rules written as 
general formulas would take the form: 

AB'^ + AC‘^ = BC'^ 

BC^- AC‘^ = AB^ 

BC^^- AB^^ AC^ 

From these formulas we have, by extracting the 
square root on each side of the equal sign: 

BC = VaB^ + AC^ 

AB = V BC'‘ - AC'‘ 

AC = V BC‘‘ - AB‘‘ 

These formulas make it possible to find the third 
side when two sides are given, no matter what the 


B 



numerical values of the length of the sides may 
be. Assume AB = 12, and BC = 20; find AC. Ac¬ 
cording to the formula: 

AC = V 20^ - 12'-* = V400 - 144 = =16. 

Assume that AB = 15 and AC = 20. Find BC. 

BC =V 15" + 20" = V 225 + 400 = = 25. 

The rules and formulas given make it possible to 
find the length of the sides in a right-angled tri¬ 
angle. To find the angles, however, use must be 















ELEMENTS OF TRIGONOMETRY 101 

made of the trigonometric functions, the meanings 
of which will be presently explained. The trigo¬ 
nometric functions are the sine, cosine, tangent, co¬ 
tangent, secant and cosecant of angles. While these 
functions are used in the solution of all kinds of 
triangles, they refer directly to right-angled tri¬ 
angles, and the meaning or value of each function 
can be explained by reference to a right-angled 
triangle as shown in Fig. 91, at G, where the side 
BC is the hypotenuse, AC the side adjacent to 
angle D, and AB the side opposite angle D, Of 
course, if reference is made to angle E, then AB 
is the side adjacent and AC the side opposite. 

The sine of an angle is the length of the opposite 
side, if the hypotenuse is assumed to equal 1. The 
sine of angle D, then, is the length of AB if BC 
equals 1. To find the sine of D when BC is any 
other length, divide AB hy the length of BC, To 
find the sine of D, if BC equals 5, for example, it 
is necessary to divide the length of AB by 5. 

Find the sine of D, when AB = 15 and BC = 20. 
The sine of D = 15 20 = 0.75. 

The cosine of an angle is the length of the adja¬ 
cent side, if the hypotenuse is assumed to equal 1. 
The cosine of angle D, then, is the length of AC 
if BC equals 1. To find the cosine of D when BC 
is any other length, divide AC hy the length of 
BC. To find the cosine of D, if BC equals 8, for 
example, it is necessary to divide the length of 
AC by 8. 

Find the cosine of D, when AC = 12 and BC = 
30. The cosine of Z) = 12 30 = 0.4. 

The tangent of an angle is the length of the op- 


102 SELF-TAUGHT MECHANICAL DRAWING 

posite side, if the adjacent side is assumed to 
equal 1. The tangent of angle D is the length of 
AB \i AC equals 1. To find the tangent of D when 
AC equals any other length, divide AB by the 
length of AC, To find the tangent of D when AC 
equals 3, for example, it is necessary to divide the 
length of AB by 3. 

Find the tangent of D, when AB = 16 and AC 
= 12. The tangent of i) = 16 12 = 1.333. 

The cotangent of an angle is the length of the 
adjacent side, if the opposite side is assumed to 
equal 1. The cotangent of angle D is the length 
of AC if AB equals 1. To find the cotangent of D 
when AB equals any other length, divide AC by 
the length of AB. To find the cotangent of D 
when AB equals 12, for example, divide AC by 12. 

Find the cotangent of D when AB = 3 and AC 
= 36. The cotangent of Z) = 36 ^ 3 = 12. 

The secant of an angle is the length of the hypo¬ 
tenuse, if the adjacent side is assumed to equal 1. 
The secant of angle D is the length of BC when 
AC equals 1. To find the secant of D when AC is 
any other length, divide BC by the length of AC. 

Find the secant of D when BC = 24 and AC = 9. 
The secant of Z) = 24 9 = 2.666. . . 

The cosecant of an angle is the length of the 
hypotenuse if the opposite side is assumed to equal 
1. The cosecant of angle D is the length of BC 
when AB equals 1. To find the cosecant of D when 
AB is any other length, divide BC by the length 
of AB. 

Find the cosecant of D when BC = 30 and AB 
= 3.75. The cosecant of Z) = 30 3.75 = 8. 


ELEMENTS OF TRIGONOMETRY 103 

The expressions sine, cosine, tangent, cotangent, 
secant and cosecant are abbreviated as follows: 
sin, cos, tan, cot, sec, and cosec. Instead of writ¬ 
ing tangent of D, for example, it is usual to write 
tan, D. By means of these functions, tables of 
which are given in the following, the values of 
angles can be introduced in the calculations of tri¬ 
angles. The tables here used give the values of 
the functions of angles for every degree and for 
every ten minutes. Only three decimal places are 
given, as that is enough for the great majority of 
shop calculations. When very accurate calculations 
are required, tables can be procured giving the 
functions for every minute, and with five decimal 
places. From the tables given, when the angle is 
known, the corresponding angular function can be 
found, and when the function is known, the cor¬ 
responding angle can be determined by merely 
reading off the values in the table. The tables in¬ 
clude sines, cosines, tangents and cotangents only, 
as these are most commonly used, and all problems 
can be solved by the use of them. When the se¬ 
cant is required, it can be found by dividing 1 by 
the cosine. The cosecant is found by dividing 1 
by the sine. 

The tables of sines, cosines, etc., are read the 
same as any other table. It will be seen that the 
four tables given are headed Sines, Cosines, Tan¬ 
gents, and Cotangents, respectively. At the bottom 
of the table headed '‘Sines'’ is read the word 
"Cosines," and at the bottom of the table headed 
"Cosines" is read the word "Sines." In the same 
way, at the bottom of the table headed "Tan- 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 


89 

88 

87 

86 

85 

84 

83 

82 

81 

80 

79 

78 

77 

76 

75 

74 

73 

72 

71 

70 

69 

68 

67 

66 

65 

64 

63 

62 

61 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 


SINES 


Minutes. 


10 ' 

20 ' 

30 ' 

40 ' 

50 ' 

0. 

003 

0. 

006 

'0. 

009 

0. 

012 

0. 

015 

0. 

020 

0. 

023 

0. 

026 

0. 

029 

0 

032 

0. 

038 

0. 

041 

0. 

044 

0. 

047 

0 

049 

0. 

055 

0. 

058 

0. 

061 

0. 

064 

0 

067 

0. 

073 

0. 

076 

0. 

078 

0. 

081 

0 

084 

0. 

090 

0. 

093 

0. 

096 

0. 

099 

0 

102 

0. 

107 

0. 

110 

0. 

113 

0. 

116 

0 

119 

0. 

125 

0. 

128 

0. 

131 

0. 

133 

0 

136 

0. 

142 

0. 

145 

0. 

148 

0. 

151 

0 

.154 

0. 

159 

0. 

162 

0. 

165 

0 

168 

0 

.171 

0. 

177 

0. 

179 

0. 

182 

0 

185 

0 

.188 

0. 

194 

0. 

197 

0. 

199 

0 

202 

0 

.205 

0. 

211 

0. 

214 

0. 

216 

0 

219 

0 

.222 

0 

228 

0. 

231 

0. 

233 

0 

236 

0 

.239 

0 

245 

0. 

248 

0 

250 

0 

253 

0 

.256 

0 

262 

0. 

264 

0 

267 

0 

270 

0 

.273 

0 

278 

0. 

281 

0 

284 

0 

287 

0 

.290 

0 

295 

0. 

298 

0 

301 

0 

303 

0 

.306 

0 

312 

0 

315 

0 

317 

0 

320 

0 

.323 

0 

328 

0 

331 

0 

334 

0 

337 

0 

.339 

0 

345 

0 

347 

0 

350 

0 

353 

0 

.356 

0 

361 

0 

364 

0 

367 

0 

369 

0 

.372 

0 

377 

0 

380 

0 

383 

0 

385 

0 

.388 

0 

.393 

0 

396 

0 

399 

0 

401 

0 

.404 

0 

.409 

0 

412 

0 

415 

0 

417 

0 

.420 

0 

.425 

0 

428 

0 

431 

0 

433 

0 

.436 

0 

.441 

0 

444 

0 

446 

0 

449 

0 

.451 

0 

.457 

0 

459 

0 

462 

0 

464 

0 

.467 

0 

.472 

0 

475 

0 

477 

0 

480 

0 

.482 

0 

.487 

0 

490 

0 

492 

0 

495 

0 

.497 

0 

.503 

0 

505 

0 

508 

0 

.510 

0 

.513 

0 

.518 

0 

520 

0 

.522 

0 

.525 

0 

.527 

0 

.532 

0 

.535 

0 

.537 

0 

.540 

0 

.542 

0 

.547 

0 

.550 

0 

.552 

0 

.554 

0 

.557 

0 

.562 

0 

.564 

0 

.566 

0 

.569 

0 

.571 

0 

.576 

0 

.578 

0 

.581 

0 

.583 

0 

.585 

0 

.590 

0 

.592 

0 

.595 

0 

.597 

0 

.599 

0 

.604 

0 

.606 

0 

.609 

0 

.611 

0 

.613 

0 

.618 

0 

.620 

0 

.623 

0 

.625 

0 

.627 

0 

.632 

0 

.634 

0 

.636 

0 

.638 

0 

.641 

0 

.645 

0 

.647 

0 

.649 

0 

.652 

0 

.654 

0 

.658 

0 

.660 

0 

.663 

0 

.665 

0 

.667 

0 

.671 

0 

.673 

0 

.676 

0 

.678 

0 

.680 

0 

.684 

0 

.686 

0 

.688 

0 

.690 

0 

.693 

0 

.697 

0 

.699 

0 

.701 

0 

.703 

0 

.705 

50 ' 

40 ' 

30 ' 

20 ' 

10 ' 


Minutes. 


60 ' 


COSINES 
















































0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 


COSINES 105 


10 ' 


1.000 

1.000 

0.999 

0.998 

0.997 

0.996 

0.994 

0.992 

0.990 

0.987 

0.984 

0.981 

0.978 

0.974 

0.970 

0.965 

0.960 

0.955 

0.950 

0.945 

0.939 

0.933 

0.926 

0.919 

0.912 

0.905 

0.898 

0.890 

0.882 

0.873 

0.865 

0.856 

0.847 

0.837 

0.827 

0.817 

0.807 

0.797 

0.786 

0.775 

0.764 

0.753 

0.741 

0.729 

0.717 


50 ' 


20 ' 


1.000 
1.000 
0.999 
0.998 
0.997 
0.996 
0.994 
0.992 
0.989 
0.987 
0.984 
0.981 
0.977 
0.973 
0.969 
0.964 
0.960 
0.955 
0.949 
0.944 
0.938 
0.931 
0.925 
0.918 
0.911 
0.904 
0.896 
0.888 
0.880 
0.872 
0.863 
0.854 
0.845 
0.835 
0.826 
0.816 
0.806 
0.795 
0.784 
0.773 
0.762 
0.751 
0.739 
0.727 
0.715 


40 ' 


30 ' 


1.000 

1.000 

0.999 

0.998 

0.997 

0.995 

0.994 

0.991 

0.989 

0.986 

0.983 

0.980 

0.976 

0.972 

0.968 

0.964 

0.959 

0.954 

0.948 

0.943 

0.937 

0.930 

0.924 

0.917 

0.910 

0.903 

0.895 

0.887 

0.879 

0.870 

0.862 

0.853 

0.843 

0.834 

0.824 

0.814 

0.804 

0.793 

0.783 

0.772 

0.760 

0.749 

0.737 

0.725 

0.713 


30 ' 


40 ' 


l.-OOO 

1.000 

0.999 

0.998 

0.997 

0.995 

0.993 

0.991 

0 . 98 ) 

0.986 

0.983 

0.979 

0.976 

0.972 

0.967 

0.963 

0:958 

0.953 

0.947 

0.942 

0.936 

0.929 

0.923 

0.916 

0.909 

0.901 

0.894 

0.886 

0.877 

0.869 

0.860 

0.851 

0.842 

0.832 

0.822 

0.812 

0.802 

0.792 

0.781 

0.770 

0.759 

0.747 

0.735 

0.723 

0.711 


20 ' 


50 ' 


1.000 

0.999 

0.999 

0.998 

0.996 

0.995 

0.993 

0.991 

0.988 

0.985 

0.982 

0.979 

0.975 

0.971 

0.967 

0.962 

0.957 

0.952 

0.946 

0.941 

0.935 

0.928 

0.922 

0.915 

0.908 

0.900 

0.892 

0.884 

0.876 

0.867 

0.859 

0.850 

0.840 

0.831 

0.821 

0.811 

0.800 

0.790 

0.779 

0.768 

0.757 

0.745 

0.733 

0.721 

0.709 


10 ' 


60 ’ 


1.000 

0.999 

0.999 

0.998 

0.996 

0.995 

0.993 

0.990 

0.988 

0.985 

0.982 

0.978 

0.974 

0.970 

0.966 

0.961 

0.956 

0.951 

0.946 

0.940 

0.934 

0.927 

0.921 

0.914 

0.906 

0.899 

0.891 

0.883 

0.875 

0.866 

0.857 

0.848 

0.839 

0.829 

0.819 

0.809 

0.799 

0.788 

0.777 

0.766 

0.755 

0.743 

0.731 

0.719 

0.707 


O' 


Deg. 


89 

88 

87 

86 

85 

84 

83 

82 

81 

80 

79 

78 

77 

76 

75 

74 

73 

72 

71 

70 

69 

68 

67 

66 

65 

64 

63 

62 

61 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 

45 


Deg. 


Minutes. 


Minutes. 


SINES 








































1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 


87 

86 

85 

84 

83 

82 

81 

80 

79 

78 

77 

76 

75 

74 

73 

72 

71 

70 

69 

68 

67 

66 

65 

64 

63 

62 

61 

60 

59 

58 

57 

56 

55 

54 

53 

52 

51 

50 

49 

48 

47 

46 


TANGENTS 


Minutes. 


0.000 
0.017 
0.035 
0.052 
0.070 
0.087 
0.105 
0.123 
0.141 
0.158 
0.176 
0.194 
0.213 
0.231 
0.249 
0.268 
0.287 
0.306 
0.325 
0.344 
0.364 
0.384 
0.404 
0.424 
0.445 
0.466 
0.488 
0.510 
0.532 
0.554 
0.577 
0.601 
0.625 
0.649 
0.675 
0.700 
0.727 
0.754 
0.781 
0.810 
0.839 
0.869 
0.900 
0.933 
0.966 


10 ' 

20 ' 

30 ' 

40 ' 

50 ' 

0 

.003 

0 

.006 

0 

.009 

0 

.012 

0 

015 

0 

.020 

0 

.023 

0 

.026 

0 

.029 

0 

.032 

0 

.038 

0 

.041 

0 

.044 

0 

.047 

0 

.049 

0 

.055 

0 

.058 

0 

.061 

0 

.064 

0 

.067 

0 

.073 

0 

.076 

0 

.079 

0 

.082 

0 

.085 

0 

.090 

0 

.093 

0 

.096 

0 

.099 

0 

.102 

.0 

.108 

0 

.111 

0 

.114 

0 

.117 

0 

.120 

0 

.126 

0 

.129 

0 

.132 

0 

.135 

0 

.138 

0 

.144 

0 

.146 

0 

.149 

0 

.152 

0 

.155 

0 

.161 

0 

.164 

0 

.167 

0 

.170 

0 

.173 

0 

.179 

0 

.182 

0 

.185 

0 

.188 

0 

.191 

0 

.197 

0 

.200 

0 

.203 

0 

.206 

0 

.210 

0 

.216 

0 

.219 

0 

.222 

0 

.225 

0 

.228 

0 

.234 

0 

.237 

0 

.240 

0 

.243 

0 

.246 

0 

.252 

0 

.256 

0 

.259 

0 

.262 

0 

.265 

0 

.271 

0 

.274 

0 

.277 

0 

.280 

0 

.284 

0 

.290 

0 

.293 

0 

.296 

0 

.299 

0 

.303 

0 

.309 

0 

.312 

0 

.315 

0 

.318 

0 

.322 

0 

.328 

0 

.331 

0 

.335 

0 

.338 

0 

.341 

0 

.348 

0 

.351 

0 

.354 

0 

.357 

0 

.361 

0 

.367 

0 

.371 

0 

.374 

0 

.377 

0 

.381 

0 

.387 

0 

.391 

0 

.394 

0 

.397 

0 

.401 

0 

.407 

0 

.411 

0 

.414 

0 

.418 

0 

.421 

0 

.428 

0 

.431 

0 

.435 

0 

.438 

0 

.442 

0 

.449 

0 

.452 

0 

.456 

0 

.459 

0 

.463 

0 

.470 

0 

.473 

0 

.477 

0 

.481 

0 

.484 

0 

.491 

0 

.495 

0 

.499 

0 

.502 

0 

.506 

0 

.513 

0 

.517 

0 

.521 

0 

.524 

0 

.528 

0 

.535 

0 

.539 

0 

.543 

0 

.547 

0 

.551 

0 

.558 

0 

.562 

0 

.566 

0 

.570 

0 

.573 

0 

.581 

0 

.585 

0 

.589 

0 

.593 

0 

.597 

0 

.605 

0 

.609 

0 

.613 

0 

.617 

0 

.621 

0 

.629 

0 

.633 

0 

.637 

0 

.641 

0 

.645 

0 

.654 

0 

.658 

0 

.662 

0 

.666 

0 

.670 

0 

.679 

0 

.683 

0 

.687 

0 

.692 

0 

.696 

0 

.705 

0 

.709 

0 

.713 

0 

.718 

0 

.722 

0 

.731 

0 

.735 

0 

.740 

0 

.744 

0 

.749 

0 

.758 

0 

.763 

0 

.767 

0 

.772 

0 

.777 

0 

.786 

0 

.791 

0 

795 

0 

.800 

0 

.805 

0 

.815 

0 

.819 

0 

.824 

0 

.829 

0 

.834 

0 

.844 

0 

849 

0 

854 

0 

859 

0 

.864 

0 

.874 

0 

880 

0 

885 

0 

890 

0 

.895 

0 

906 

0 

.911 

0 

916 

0 

922 

0 

.927 

0 

9.38 

0 

.943 

0 

949 

0 

955 

0 

.960 

0 

.971 

0 

.977 

0 

983 

0 

988 

0 

994 

50 ' 

40 ' 

30 ' 

20 ' 

10 ' 


Minutes. 


COTANGENl'S 








































COTANGENTS 


107 


Minutes. 


Oeg 


Deg . 


O ' 

10 ' 

20 ' 

30 ' 

40 ' 

50 ' 

60 ' 


0 

oo 

343.8 

171.9 

114.6 

85.94 

68.75 

57.29 

89 

1 

57.29 

49.10 

42.96 

38.19 

34.37 

31.24 

28.64 

88 

2 

28.64 

26.43 

24.54 

22.90 

21.47 

20.21 

19.08 

87 

3 

19.08 

l 8.07 

17.17 

16.35 

15.60 

14.92 

14.30 

86 

4 

14.30 

l 3.73 

13.20 

12.71 

12.25 

11.83 

11.43 

85 

5 

11.43 

11.06 

10.71 

10.39 

10.08 

9.788 

9.514 

84 

0 

9.514 

9.225 

9.010 

8.777 

8.556 

8.345 

8.144 

83 

7 

8.144 

7.953 

7.770 

7.596 

7.429 

7.269 

7.115 

82 

8 

7.115 

6.968 

6.827 

6.691 

6.561 

6.435 

6.314 

81 

9 

6.314 

6.197 

6.084 

5.976 

5.871 

5.769 

5.671 

80 

10 

5.671 

5.576 

5.485 

5.396 

5.309 

5.226 

5.145 

79 

11 

5.145 

5.066 

4.989 

4.915 

4.843 

4.773 

4.705 

78 

12 

4.705 

4.638 

4.574 

4.511 

4.449 

4.390 

4.331 

77 

13 

4.331 

4.275 

4.219 

4.165 

4.113 

4.061 

4.011 

76 

14 

4.011 

3.962 

3.914 

3.867 

3.821 

3.776 

3.732 

75 

15 

3.732 

3.689 

3.647 

3.606 

3.566 

3.526 

3.487 

74 

16 

3.487 

3.450 

3.412 

3.376 

3.340 

3.305 

3.271 

73 

17 

3.271 

3.237 

3.204 

3.172 

3.140 

3.108 

3.078 

72 

18 

3.078 

3.047 

3.018 

2.989 

2.960 

2.932 

2.904 

71 

19 

2.904 

2.877 

2.850 

2.824 

2.798 

2.773 

2.747 

70 

20 

2.747 

2.723 

2.699 

2.675 

2.651 

2.628 

2.605 

69 

21 

2.605 

2.583 

2.560 

2.539 

2.517 

2.496 

2.475 

68 

22 

2.475 

2.455 

2.434 

2.414 

2.394 

2.375 

2.356 

67 

23 

2.356 

2.337 

2.318 

2.300 

2.282 

2.264 

2.246 

66 

24 

2.246 

2.229 

2.211 

2.194 

2.177 

2.161 

2.145 

65 

25 

2.145 

2.128 

2.112 

2.097 

2.081 

2.066 

2.050 

64 

26 

2.050 

2.035 

2.020 

2.006 

1.991 

1.977 

1.963 

63 

27 

1.963 

1.949 

1.935 

1.921 

1.907 

1.894 

1.881 

62 

28 

1.881 

1.868 

1.855 

1.842 

1.829 

1.816 

1.804 

61 

29 

1.804 

1.792 

1.780 

1.767 

1.756 

1.744 

1.732 

60 

30 

1.732 

1.720 

1.709 

1.698 

1.686 

1.675 

1.664 

59 

31 

1.664 

1.653 

1.643 

1.632 

1.621 

1.611 

1.600 

58 

32 

1.600 

1.590 

1.580 

1.570 

1.560 

1.550 

1.540 

57 

33 

1.540 

1.530 

1.520 

1.511 

1.501 

1.492 

1.483 

56 

34 

1.483 

1.473 

1.464 

1.455 

1.446 

1.437 

1.428 

55 

35 

1.428 

1.419 

1.411 

1.402 

1.393 

1.385 

1.376 

54 

36 

1.376 

1.368 

1.360 

1.351 

1.343 

1.335 

1.327 

53 

37 

1.327 

1.319 

1.311 

1.303 

1.295 

1.288 

1.280 

52 

38 

1.280 

1.272 

1.265 

1.257 

1.250 

1.242 

1.235 

o 1 

39 

1.235 

1.228 

1.220 

1.213 

1.206 

1.199 

1.192 

50 

40 

1.192 

1.185 

1.178 

1.171 

1.164 

1.157 

1.150 

49 

A o 

41 

1.150 

1.144 

1.137 

1.130 

1.124 

1.117 

1.111 

48 

A 7 

42 

1.111 

1.104 

1.098 

1.091 

1.085 

1.079 

1.072 

47 

4 r * 

43 

1.072 

1.066 

1.060 

1.054 

1.048 

1.042 

1.036 

46 

A n 

44 

1.036 

1.030 

1.024 

1.018 

1.012 

1.006 

1.000 

4 o 


60 ' 

50 ' 

40 ' 

30 ' 

20 ' 

10 ' 

O ' 

Deg . 

Eeg 











Minutes. 






TAN(iEN rS 




























































108 SELF-TAUGHT MECHANICAL DRAWING 

gents,we read ‘‘Cotangents,’’ and at the bottom 
of the table headed “Cotangents,” we read “Tan¬ 
gents.” The object of this will be presently ex¬ 
plained. The extreme left-hand column, we find, is 
headed “Deg.,” and the following seven columns 
are headed O', 10', 20', 30', 40', 50' and 60', re¬ 
spectively, these columns indicating the minutes. 
At the bottom of the pages the same numbers are 
found but reading from the right to the left. The 
values of the functions marked at the top are read 
in the table opposite the degrees in the left-hand 
column and under the minutes at top. The values 
of the functions marked at the bottom are read 
opposite the degrees in the right-hand column and 
over the minutes at the bottom. For example, the 
sine of 39° 40' or sin 39° 40', as it is written in 
formulas, is thus found to be 0.638, and the sine 
of 64° 10' is 0.900, this latter value being read off 
in the second table, reading it from the bottom up, 
and locating the number of degrees in the right- 
hand column. 

As further examples, we find 

tan 37° 40' = 0.772 
cot 37° 40' = 1.295 
tan80° 0' = 5.671 
cos 75° 30' = 0.250 

We are now ready to proceed to solve right-ang¬ 
led triangles with regard both to the sides and the 
angles. In any right-angled triangle, if either two 
sides, or one side and one of the acute angles are 
known, the remaining quantities can be found. As 
a general rule, in any triangle, all the quantities 


OPPOSITE-SIDE- 


ELEMENTS OF TRIGONOMETRY 109 

can be found when three quantities, at least one 
of which is a side, are given. In a right-angled 
triangle the right angle is always known, of 
course, so that here, therefore, only two additional 
quantities are necessary. If all the three angles 
are known, the length of the sides cannot be de¬ 
termined ; one side, at least, must also always be 
known in order to make possible the solution of 
the triangle. 

The following rules should be used for solving 
right-angled triangles. 

Case 1. Two sides known.—Use the rules al¬ 
ready given in this chapter for finding the third 



side when two sides in a right-angled triangle are 
given. To find the angles use the rules already 
given for finding sines, cosines, etc., and the 
tables. 

Case 2. Hypotenuse and one angle given.—Call 
the side adjacent to the given angle the adjacent 
side, and the side opposite the given angle the 
opposite side (see Fig. 92.) Then the adjacent 
side equals the hypotenuse multiplied by the cosine 












110 SELF-TAUGHT MECHANICAL DRAWING 

of the given angle; the opposite side equals the 
hypotenuse multiplied by the sine of the given 
angle; and the unknown angle equals 90 degrees 
minus the given angle. 

Case 3. One angle and its adjacent side given. 
—The hypotenuse equals the adjacent side divided 
by the cosine of the given angle; the opposite side 
equals the adjacent side multiplied by the tangent 
of the given angle; and the unknown angle is 
found as in Case 2. 

Case Jf. One angle and its opposite side known. 
—The hypotenuse equals the opposite side divided 
by the sine of the given angle; the adjacent side 
equals the opposite side multiplied by the cotangent 
of the given angle; and the unknown angle is 
found as in Case 2. 

These rules may be written as formulas as fol¬ 
lows (see Fig. 93): 

Case 1. For formulas for the sides see the first 
part of this Chapter. For the angles we have: 

• Ty b . yry C 

Sin B = — sin C = —. 

a a 

Case 2. Here, when a and B are given, we have: 
c = a cos B; b = a sin B; C = 90 ° — B. 

When a and C are given, we have: 

b = a cos C; c = a sin C; 5 = 90 ° — C. 

Case 3. Here, when B and care given, we have: 

' a = ^ - b = c tan i?; C = 90 ° — B, 

cos B ' 

When C and b are given, we have: 

a = c = b tan C; = 90 ° — C. 

cos c 



ELEMENTS OF TRIGONOMETRY 


111 


Case Jf, H6r6, when B and 6 ar© known, W6 
have: 

^ "" sirTS’ ^ ^ cot B; C = 90° - B. 

When C and c are known, we have: 

« = 6 = c cot C; 5 = 90° - C. 

These rules and formulas, while not including all 
possible combinations for the solution of right- 
angled triangles, give all the information neces¬ 
sary for the solution of any kind of a right-angled 




triangle. A few examples of the use of these rules 
and formulas will now be given, so as to clearly 
indicate the mode of procedure in practical work. 

Example 1 .—In the triangle in Fig. 94, side AC 
is 12 inches long and angle D is 40 degrees. Find 
angle E and the two unknown sides. 

This is an example of Case 3, one angle and its’ 
adjacent side being given. Angle E equals 90 de¬ 
grees minus the given angle, or 


^7= 90°- 40° = 50° 









112 SELF-TAUGHT MECHANICAL DRAWING 


The hypotenuse BC equals the adjacent side 
divided by the cosine of D, or 

12 12 

BC = o = 15.666 inches, 

cos 40 0.766 

Side AB equals the adjacent side multiplied by 
the tangent of D, or 

AB = 12 X tan 40° = 12 X 0.839 = 10.068 inches. 

The cosine and tangent of 40 degrees are found 
in the tables of trigonometric functions as already 
explained. 

Example 2, —In the triangle in Fig. 95, the 
hypotenuse BC = 17J inches. One angle is 44 de¬ 
grees. Find angle E and the sides AB and AC. 

This is an example of Case 2, the hypotenuse 
and one angle being given. Using the rules or 
formulas given for Case 2, we have: 

AC = 17i X cos 44° = 17.5 X 0.719 = 12.5825 
inches. 

AB = 17i X sin 44° = 17.5 X 0.695 = 12.1625 
inches. 

^7= 90°- 44° =46°. 

Example 3. —In the triangle in Fig. 96, side 
= 208 feet, and the angle opposite this side = 38 
degrees. Find angle E, and the two remaining 
sides. 

This is an example of Case 4, one side and the 
angle opposite it being known. From the rules or 
formulas given for Case 4, we have: 

BC = 208 - sin 38° = 208 - 0.616 = 337.66 feet. 

AB = 208 X cot 38° = 208 X 1.280 = 266.24 feet. 

F;= 90°-38° = 52°. 




ELEMENTS OF TRIGONOMETRY 


113 


Example In the triangle in Fig. 97, side AC 
= 3 inches, and the hypotenuse BC=b inches. Find 
side AB and angles D and E. 

This is an example of Case 1. According to a 
formula pre viously giv en in this chapter 

^ ^ - AC^ = V5 - - 3 = V25 - 9 = 

\/l6 = 4. 


sin E 


AB 

BC 


4 

5 


= 0.800. 


From the tables we find that the angle corre- 


B 




spending to a sine which equals 0.800 is 53° 10'. 
Consequently: 

F7=53°10 ', and 71 = 90°-53° 10'= 36° 50', 

Example 5 .—In the triangle in Fig. 98, side BC, 
the hypotenuse, is If inch long. One angle is 65 
degrees. Find angle E and the remaining sides. 














114 SELF-TAUGHT MECHANICAL DRAWING 

This is an example of Case 2. We have: 
£'=90°- 65° =25°. 

AB=1%X cos 65° = 1.375 X 0.423 = 0.5816 inch! 
XLC = If X sin 65° = 1.375 X 0.906 =1.2457 inch. 

Example 6 .—In the triangle in Fig. 99, side AB 
= 0.706 inch, and the angle adjacent to this side is 
60 degrees. Find angle £and the sides AC and J5C. 


A B 



£ = 90°- 60° = 30°. 

BC = 0.706 - cos 60° = 0.706 - 0.500 = 1.412 
inch. 

AC= 0.706 X tan 60° = 0.706 X 1.732= 1.2228 
inch. 

The previous examples, carefully studied, will 
give a comprehensive idea of the methods used for 
solving right-angled triangles, no matter which 
parts are given or unknown. 

A triangle which does not contain a right angle 
is called an oblique triangle. Any such triangle 
can be solved by the aid of the formulas given for 
the right triangle, by dividing it into two right- 
angled triangles by means of a line drawn from 
the vertex of one angle perpendicular towards the 
opposite side. Formulas can be deduced which do 





ELEMENTS OF TRIGONOMETRY 


115 


not require that the triangle be so divided, but for 
elementary purposes, the method indicated is the 
most easily understood. 

In Fig. 100, for example, a triangle is given as 
shown. One angle is 50 degrees, and the sides in¬ 
cluding this angle are 4 and 5 inches long, respec¬ 
tively. Draw a line from A perpendicular to the 



side BC. We have here two right-angled tri¬ 
angles, and can now proceed by using the formulas 
previously given. In triangle ADB, the hypoten¬ 
use AB and one angle are given. We then find 
side AD by means of the formulas for Case 2, and 
also angle BAD and side BD. Next we find CD 
= 5- BD. We then, in the triangle ACD know 
two sides AD and CD, and can thus find side AC 
as in Case 1, as well as angles ACD and CAD. 
The angle BAC finally is found by adding angles 











116 SELF-TAUGHT MECHANICAL DRAWING 


BAD and CAD and, then, all the angles and sides 
in the triangle are found. 

The successive calculations would be carried out 
as follows: 

Ail = 4 X sin 50° = 4 X 0.766 = 3.064. 

= 4 X cos 50°= 4 X 0.643 = 2.572. 

Angle BAD = 90° - 50° = 40°. 

DC 5- BD = 5 - 2.572 = 2.428. 

AC==VaD'^ + DC' = V3.064‘' + 2.428' = 3.91. 

Sine of angle ACD = 0.784. 

Angle A CZ) = 51 ° 40 

Angle CAD = 90°- 51° 40' = 38° 20'. 

Angle 5A C = 40 ° + 38 ° 20' = 78 ° 20'. 

In order to check the results obtained, add angles 
ABC, BAC and ACD. The sum of these angles 
must equal 180 degrees if the results are correct: 

50° + 78° 20' + 51° 40' = 180°. 


This method, with such modifications as are 
necessary to meet the different requirements in 
each problem, may be used for solving all oblique- 
angled triangles, except in the case where no angle 
is known, but only the lengths of all the three 
sides. In this case the use of a direct formula 
will prove the best and most convenient. Let the 
three known sides be a, b and c, and the angles 
opposite each of them A, B and C, respectively, as 
in Fig. 101; then we have: 


b- + c 


cos A „ , 

2 be 

C = 180°- (A + B), 


a' . ^ b sin A 


; sin B = 


a 







ELEMENTS OF TRIGONOMETRY 


117 


As an example, assume that the three sides in a 
triangle are a = 4, b — 5, and c = 6 inches long. 
Find the angles. 


Cos A = 


5 ^ + 

2X5X6 


45 

60 


0.750. 


Sin B 


A = 41° 25'. 

5 X sin 41 ° 25' _ 5 X 0.662 
4 4 


0.827. 


B = 55° 50'. 


C= 180° - (41° 25' + 55° 50') = 82° 45'. 


As only the first principles of trigonometry have 
here been treated, some of the more advanced 



problems have, by necessity, been omitted. For 
ordinary shop calculations the present treatment 
will, however, be found more satisfactory, as some 
of the matter which would unnecessarily burden 
the mind has been left out. If the student only 
first acquires a thorough understanding of the first 








118 SELF-TAUGHT MECHANICAL DRAWING 


principles of mathematics and their application to 
machine design, it is comparatively easy to broaden 
the field of one’s knowledge; it is, therefore, of 
extreme importance that these first principles 
be thoroughly understood and digested. The ap¬ 
plication will then be found comparatively easy. 

The trigonometric functions afford a convenient 
means for laying out angles; and when the sides 



'Fig. 102.—Method of Laying Out Angles by Means of 

Natural Functions. 


of the angle laid out are much extended, it can 
be laid out more accurately in this manner than 
by the use of an ordinary protractor. Let it be 
required, for instance, to lay out an angle of 37 
degrees, one side of the angle being 60 inches long. 
Lay out the side AB, Fig. 102, 60 inches long. 
Then with a radius equal to the sine of 37 degrees 
multiplied by 60, and with a center at B, draw an 













ELEMENTS OF TRIGONOMETRY 


119 


arc C. Then draw a line from A, tangent to arc 
C. This line forms an angle of 37 degrees with 
line AB. If the required angle is over 45 degrees, 
then it is preferable to lay out the complement 
angle from a line perpendicular to the original 



line, as shown in Fig. 103, where an angle of 70 
degrees is to be laid out, but the 20-degree comple¬ 
ment angle is actually constructed. Many other 
methods for use in laying out angles, arcs, etc., 
will readily suggest themselves to the student who 
thoroughly understands the relation of the trigo¬ 
nometric functions in a right-angled triangle. 





CHAPTER VIII 

ELEMENTS OF MECHANICS 

Mechanics is defined as that science, or branch 
of applied mathematics, which treats of the action 
of forces on bodies. That part of mechanics which 
considers the action of forces in producing rest or 
equilibrium is called statics; that which relates to 
such action in producing motion is called dynamics; 
the term mechanics includes the action of forces 
on all bodies whether solid, liquid or gaseous. It 
is sometimes, however, and formerly was ’often, 
used distinctively of solid bodies only. The me¬ 
chanics of liquid bodies is called also hydrostatics 
or hydrodynamics, according as the laws of rest or 
motion are considered. The mechanics of gaseous 
bodies is called also pneumatics. The mechanics 
' of fluids in motion, with special reference to the 
methods of obtaining from them useful results, 
constitutes hydraulics. 

The Resultant of Two or More Forces.—When a 
body is acted upon by several forces of different 
magnitudes in different directions, a single force 
may be found, which in direction and magnitude 
will be a resultant of the action of the several 
forces. The magnitude and direction of this single 
force may be obtained by what is known as the 
parallelogram of forces. Let A and B, Fig. 104, 

120 


ELEMENTS OF MECHANICS 


121 


represent the direction of two forces acting simul¬ 
taneously upon P, and let their lengths represent 
the relative magnitude of the forces; then, to find 
a force which in direction and magnitude shall 
be a resultant of these two forces, draw the line C 
parallel with B, and draw the line D parallel with 
A. A diagonal of the parallelogram thus formed, 
drawn from Pto P, will give the direction, and its 



Fig. 104.— Parallelogram of Forces. 


length as compared with A and P, the relative 
magnitude, of the required force. 

That this is so m.ay be seen by considering the 
two forces as acting separately upon P. Let A be 
considered as acting upon P to move it through 
a distance equal to its length. Then P would be 
moved to F. If the force B is now caused to act 
upon P to move it through a distance equal to its 
length, P will arrive at G. As FP has the samxe 
length and direction as A, and as GPhas the same 
length and direction as B, the distance from G to 
P would be the same as the distance from P to P; 
therefore, PP, the diagonal of the parallelogram 
formed by the lines A, B, C, and D, represents the 
required new force or resultant. 

If there are more than two forces acting upon 
the point P, first find a resultant of any two of the 
forces; then consider this resultant as replacing 



122 SELF-TAUGHT MECHANICAL DRAWING 


the first two, and find the resultant of it and an¬ 
other of the original forces; continue this process 
until a force is obtained which will be the resultant 
of all of the original forces. Thus, in Fig. 105, if 
Ay B and C be considered as representing in di¬ 


rection and magnitude 
three forces which are 
acting simultaneously 
uponP; then, if we draw 
a parallelogram upon A 
and B, we have its diag¬ 
onal PD as the resultant 
of A and B. A parallel¬ 
ogram is now drawn 



Fig. 105.—Resultant of Three 
Forces. 


upon PD and C, giving PE, its diagonal, as the 
resultant of these two, and, consequently, of the 
three original forces. 

This principle holds true whether the original 
forces are acting in the same plane or not. Thus, 
in Fig. 106, let A, B 
and C be three forces 



acting simultaneously 
upon P. Then the re¬ 
sultant of A and B 


would be the diagonal 

PD. Considering this 106.—Resultant of Three 

as replacing A and B, Forces in Different Planes, 
a resultant of it and C 

would be a diagonal drawn from P to the further 
corner E; PE would then be the resultant of A, B 
and C. 

This operation may, of course, be reversed to 
allow of finding two or more forces in different 










ELEMENTS OF MECHANICS 


123 


directions which in magnitude shall be equivalent 
to a single known force. Thus in Fig. 107^ if PA 
represents the direction and magnitude of a given 
force which it is desired to replace by two others 
acting in the direction of PB and PC, respectively, 
then draw a line from A to PB parallel with PC, 
and draw another from A to PC parallel with PB. 
The lengths Pa and Pb g 

thus determined will 
represent the relative 
magnitudes, as com- 

pared with PA, of the , 

required new forces. 

Parallel Forces.—Let 
A and B, Fig. 108, 
represent the direction ^ 

and magnitude of two Fig. 107.— Resolution of Forces, 
parallel forces acting 

together upon the bar BE. These two forces may 
be replaced or counterbalanced by a single force, 
equal in magnitude to A and B combined. To de¬ 
termine the point of application of this new force 
produce A to a, making Da equal in length to B. 
Also make hE equal in length to A. The inter¬ 
section of the line connecting a and h with BE, at 
F, will be the required point of application. The 
lengths BE and EE will be inversely proportional 
to the forces A and B. That is, the length EE will 
be to the force A as the length BE is to the force 
B. The product of BE multiplied by A will be 
equal to the product of EE multiplied by B. 

Fig.' 109 shows how several parallel forces, act¬ 
ing in the same direction, may be replaced or 



124 SELF-TAUGHT MECHANICAL DRAWING 


counterbalanced by a single force. Let A, B and 
C represent the relative magnitudes of the forces. 
A resultant of B and C would be D, equal in 



Fig. 108.—Parallel Forces, Several Parallel Forces. 


magnitude to B and C combined, and its point of 
application, determined in the manner previously 
described, would be at a. Regarding D as r single 

force replacing B and C, 
would give E, equal in 
magnitude to A and D 
combined, as the result¬ 
ant of these two, and its 
point of application, de¬ 
termined as before, would 
be at 6. 

Oblique Forces.—Let A 

Fig. 110. —Oblique Forces j -nc 
Acting at Different Points Fig. 110, repre- 

on a Bar. Sent the directions and 

relative magnitudes of 
two forces acting simultaneously upon the bar DE. 
These two forces may be either replaced or counter- 














ELEMENTS OF MECHANICS 


125 


balanced by a single force, which in direction and 
magnitude shall be a resultant of them. Produce 
A and B until they meet at a. Draw the parallel¬ 
ogram abed, making da equal to A, and ba equal 
to B, The diagonal of this parallelogram will give 
the direction and relative magnitude of the new 
force, and if extended its intersection with DE 
will give the point of application. 

Opposing Forces.—Let A and B, Fig. Ill, repre¬ 
sent the directions and relative magnitudes of two 
forces acting upon oppo¬ 
site sides of the bar DE. 

These two forces may be 
replaced by a single force, 
which in direction and 
magnitude will be a re¬ 
sultant of them. Produce 
A and B until they meet 
at a. Lay off ac equal to 
the length of B, and make 
be equal to and parallel 
with A. A line drawn 
from a to 5 will give the 
direction of the new force, and the length of ab, 
as compared with A and B will give its relative 
magnitude. Its application on bar DE may be de¬ 
termined by extending ab until it intersects DE. 

Levers.—When a workman wishes to raise a 
heavy object, he may insert one end of a bar un¬ 
der it, and lift on the other end; or, pushing a 
block of wood or iron in under the bar as close to 
the object to be raised as he can, he presses down 
upon the free end of the bar. A bar so used con- 


ii 



Fig. 111.— Opposing Oblique 


Forces. 



126 SELF-TAUGHT MECHANICAL DRAWING 


stitutes a lever, and the point where the bar rests 
when the lever is doing its work, the end of the 
bar in under the heavy object in the first case, or 
the block on which the bar rests in the second 
case, is the fulcrum of the lever. 

Levers are of three kinds, as shown in Fig. 112: 
First, where the fulcrum is between the power 





and the weight; second, where the weight is 
between the fulcrum and the power; and, third, 
where the power is between the fulcrum and the 
weight. A man’s forearm furnishes a good illus¬ 
tration of a lever of the third class, the fulcrum 
being at the elbow, the weight at the hand, and 
the muscle, being attached to the bone of the arm, 
at a short distance from the elbow, furnishing the 
power. 















ELEMENTS OF MECHANICS 


127 


In all of these cases the gain in power is exactly 
proportional to the loss in speed, or the gain in 
speed is exactly proportional to the loss in power. 
Also, in every case the product of the weight mul¬ 
tiplied by its distance from the fulcrum, will equal 
the product of the power multiplied by its distance 
from the fulcrum, or, the weight and power will 
balance each other when the weight multiplied by 
the distance through which it moves, equals the 
power multiplied by the distance through which it 
moves. 

If in Fig. 108 the bar DE is a lever, the fulcrum 
will be at F, and the methods used in that figure 
and in Figs. 109, 110 and 111 give solutions of dif¬ 
ferent lever problems. 

The length of the lever arm is independent of 
the form of the lever. In Fig. 113 is shown a lever 



of curved shape; but the lever arms on which the 
calculation as to the work that the lever is doing, 
will be based, will be straight lines connecting the 
point where the power is applied, or the point 
which supports the weight, with the fulcrum. 

The length of the lever arm is always at right 
angles to the direction in which the power is being 








128 SELF-TAUGHT MECHANICAL DRAWING 


applied, or to the direction of the resistance of the 
weight or load. 

In Fig. 114 two cases are shown where the power 
is applied obliquely on the lever; but the lever arm 
on which the calculation is based will be the dis- 




Fig. 114.—Power Applied Obliquely on Lever. 


tance Fa measured from the fulcrum, at right 
angles to the direction of the power. 

Compound Levers.—In Fig. 115 is shown a case 
where the power gained with one lever is further 
increased by the use of a second lever, acting on 
the first one. The weight and power will balance 











ELEMENTS OF MECHANICS 


129 


each other when the product of the weight and the 
lever arms ah and ef, multiplied together, equals 
the product of the power and the lever arms gf and 
he multiplied together. Thus, to find the weight 


ah c 



which a given power will lift, divide the product 
of the power and its lever arms gf and he, multi¬ 
plied together, by the product of the lever arms of 
the weight, ah and ef, multiplied together. To find 



Fig. 116.—Diagram for Lever Problem. 

the power necessary to lift a given weight, divide 
the product of the weight and its lever arms, ah 
and ef, multiplied together, by the product of the 
lever arms of the power, gf and he, multiplied 
together. 

















130 SELF-TAUGHT MECHANICAL DRAWING 


A few examples will illustrate these principles. 
Assume that in Fig. 116 a weight at A must bal¬ 
ance the 18-pound weight at B. The lever arms 
are given as 12 and 5 inches, respectively. How 
much must the weight W be, in order to balance 
the weight at B ? 

The weight at B (18 pounds) times its lever arm 
(5 inches) must equal the weight W times its lever 
arm (12 inches). In other words: 


18 X 5 = IF X 12. 

90 = 12 W. 

W = ^ = 7i pounds. 


In Fig. 117, two weights, 4 and 2 pounds, respec¬ 
tively, are balanced by a weight W. Find what 



the weight of W must be with the lever arms 
given in the engraving. 

In this case the weight at A times its lever arm 
plus the weight at B times its lever arm, will 
equal weight W times its lever arm. The sum of 
the products of the weights and leverages of the 
weight at A and B is taken, because both these 
weights are on the same side of the fulcrum F, 


















ELEMENTS OF MECHANICS 131 


Carrying out the calculation outlined above, we 

M 5^ VP * 

4X 16+ 2X8 = 6 W, 

64 + 16 = 80 = 6 T7. 

IT = ^ = 13J pounds. 


The product of a weight or force and its lever 
arm is commonly called the moment of the force. 
The moment of the force at A, for example, is 4 
pounds X16 inches = 64 inch-pounds. If the lever 
arm were 16 feet instead of 16 inches, the result 
would be ^4: foot-pounds. 

An interesting application of the lever, and the 
moments of forces, is presented in calculations of 



weights for safety valves. A diagrammatical 
sketch of a safety valve lever is shown in Fig. 118. 
Assume that the total steam pressure, acting on 
the whole area of the safety valve, is 300 pounds 
when it is required that the steam should “blow 
off.^^ Find the weight IF required near the end 
of the lever to keep the valve down until the total 
pressure is 300 pounds on the valve. Assume the 
weight of the lever itself to be 6 pounds, con¬ 
centrated at its center of gravity, 10 inches from 
the fulcrum F, 




















132 SELF-TAUGHT MECHANICAL DRAWING 


In this case we have that the moment of the 
steam pressure, which acts upward, should equal 
the sum of the moments of the weight of the lever 
and the weight W. Therefore: 


300 X 3 = 6 X 10 + 20 W. 
900 = 60 + 20 W. 

900 - 60 = 20 W. 

840 = 20 W. 

840 
20 


W = 


42 pounds. 


The calculation above has been carried out step 
by step, so that students unfamiliar with the alge¬ 
braic solution of equations may be able to under¬ 
stand the principles involved in simple examples 
of this kind. In the following, the calculations 
have been carried out more directly, but the stu¬ 
dent should use the ‘‘step by step’^ method until 
thoroughly familiar with the subject. 

Fixed and Movable Pulleys.—A fixed pulley is 
frequently used to change the direction of the 
power, as shown in Fig. 119, but there is no gain 
in power with such a pulley, as there is no com¬ 
pensating loss of speed; the weight will move up¬ 
ward at the same rate of speed as the power moves 
downward. 

If now a movable pulley be used in connection 
with the fixed pulley as shown in Fig. 120, then as 
the end of the rope to which the power is applied 
is drawn downward, each of the two strands of 
rope between the pulleys will take half of the 
stress of the suspended weight, and the weight 
will be raised only one-half the distance that the 


ELEMENTS OF MECHANICS 


133 


power descends. The power will therefore need to 
be only one-half of the weight. In Fig. 121, there 
are three strands of rope between the pulleys, each 
of which will be equally shortened when the free 
end of the rope is pulled; the power, therefore, is 
only one-third of the weight. In Fig. 122, with 



Fig. 119.— Fixed Pulley. 



Fig. 120.— Fixed and Movable 
Pulleys. 


four strands of rope between the pulleys, each fur¬ 
nishing an equal amount to the free end as it is 
drawn out, the power need be only one-fourth of 
the weight. 

The law of the pulley, then, where a single rope 
is employed, is that the power will be increased as 
many times as there are lines of rope between the 
pulleys to participate in the shortening. In a sys¬ 
tem using more than one rope, as shown in Fig. 















134 SELF-TAUGHT MECHANICAL DRAWING 


123, each additional movable pulley doubles the 
power, as it will move at only half the rate of the 
preceding pulley. 

Differential Pulleys.—Another form of pulley, 
known as the differential pulley, much used in ma¬ 
chine shops, is shown in Fig. 124. In this form of 




Fig. 121.—Tackle where Load Fig. 122.—Tackle where Load 
is Taken on Three Strands is Taken on Four Strands 
of Rope. of Rope. 

pulley an endless chain replaces the rope, the pul¬ 
leys themselves being grooved and toothed like 
sprocket wheels. The two pulleys at the top are 
of slightly different diameters, but rotate together 
as one piece. In operation, as the chain is drawn 
up by the large wheel it passes around in a loop to 
the small wheel from which it is unwound, causing 
the loop in which the movable pulley rests to be 




















ELEMENTS OF MECHANICS 


135 


shortened by an amount equal to the difference in 
the pitch circumferences of the two upper wheels, 
when they have made one revolution. This would 
cause the weight to be raised one-half of that 
amount. If in a given case the two upper pulleys 
had respectively 20 and 19 teeth, then as the ap- 



FiG. 123.—A Special Arrange¬ 
ment of Movable Pulleys. 



Fig. 124.—Differential 
Pulley. 


plied power was being moved through a distance 
of 20 inches the small pulley would unwind 19 
inches of the chain, causing a shortening of the 
loop in which the movable pulley rests of one inch, 
which would raise the weight one-half of an inch, 
giving a ratio of load to power of 40 to 1. 

In all of these cases the results actually attained 
in practice will be somewhat modified from the 
























136 SELF-TAUGHT MECHANICAL DRAWING 

theoretical results given by calculations, by the 
losses occasioned by friction. 

Inclined Planes.—In raising heavy weights 
through short distances, as for instance in loading 
barrels onto wagons, a plank may be used to facili¬ 
tate the work by placing one end of it on the 
ground and the other end on the wagon, and roll¬ 
ing the barrel up the plank onto the wagon. Such 
an arrangement is called an inclined plane. When 
the force which is being applied to the rolling 




Fig. 125. — Inclined Plane. Fig. 126. — Power Applied 

Parallel to Base. 

object is exerted in a direction parallel to the in¬ 
clined surface, as in Fig. 125, it is evident that the 
power must move through a distance equal to the 
length of the incline in order to raise the weight 
the desired height.. The gain in power will then 
be equal to the length of the incline divided by the 
height. 

If the power is applied in a direction parallel 
with the base, as in Fig. 126, the power will have 
to advance through a distance equal to the length 
of the base to raise the object the desired height. 
The gain in power will then be equal to the base 
divided by the height. By considering Fig. 126 







ELEMENTS OF MECHANICS 


137 



d 


Fig. 127.—Power Applied Obliquely 
to Surface of Incline. 


further, it will be seen that in rolling the object 
up the incline the power will have to advance from 
the beginning of the 
incline to a point 
from which a line 
may be drawn per¬ 
pendicular to its di¬ 
rection to the top of 
the incline. In any 
case where the 
power is applied in 
any direction other 
than parallel with 
the incline, in roll¬ 
ing the object to the top, the power will have to 
advance to a point from which a line may be drawn 
perpendicularly to its direction to the top of the 

incline. In Figs. 127 
and 128 are shown two 
other cases where the 
power is applied in 
a direction obliquely 
to the surface of the 
incline. In either of 
these cases, as in the 
other two cases, the 
gain in power will 
be found by dividing 
the distance through 
which the force 
moves, ah, by the distance through which the 
object is raised, cd. 

It will be further seen that the gain in power is 



Fig. 128. —Another Case where 
Power is Applied Obliquely to 
Surface of Incline. 






138 SELF-TAUGHT MECHANICAL DRAWING 


greatest when the direction in which the force is 
being applied is parallel with the incline. When 
the direction of the force is upward from the in¬ 
cline, as in Fig. 127, part of the force is expended 
in lifting the weight off from the incline, until, 
when its direction is made vertical, it is all 
expended in this way. When the direction of the 
force is downward from the incline, as in Figs. 126 
and 128, part of it is lost in pressing the object 
against the incline. 

The Screw.—The screw is a modified form of 
inclined plane, the lead of the screw, the distance 


B 


Fig. 129.—Differential Screw. 



that the thread advances in going around the 
screw once, being the height of the incline, and 
the distance around the screw, measured on the 
thread, being the length of the incline. 

The Differential Screw.—The differential screw 
is a compound screw having a coarse thread part of 
its length, and a somewhat finer thread the rest of 
its length, the object being to get a slow motion 
combined with the strength of a coarse thread. 
Fig. 129 shows such a screw. The piece A is a 
fixed part of some machine. The piston B slides 
within A,being prevented from turning by the pin 
C which enters a groove m B. If that part of the 





















































ELEMENTS OF MECHANICS 


189 


screw which engages in A has eight threads to 
the inch, and that part of it which engages in B 
has ten threads, then when the screw makes one 
revolution, it will advance into A one-eighth of an 
inch, and into B one-tenth of an inch; the piston 
B will therefore advance through a distance equal 
to the difference between one-eighth of an inch 
and one-tenth of an inch, or twenty-five one-thou¬ 
sandths of an inch, requiring forty turns of the 
screw to make the piston advance one inch. 

Newton's Laws of Motion.—The relation which 
exists between force and motion is stated by the 
three fundamental laws of motion formulated by 
Newton. 

Newton's first law says that if a body is at 
rest it will remain at rest, or, if it is in motion, it 
will continue to move at a uniform velocity in a 
straight line, until acted upon by some force and 
compelled to change its state of rest or of straight- 
line uniform motion. In a general way, this law is 
self-evident, and based on daily experience. How¬ 
ever, the part of the law stating that a body in 
motion will continue indefinitely to move if not 
acted upon by resisting forces, may not be so self- 
evident; yet whenever a body is brought to a stand¬ 
still after it has been in motion, such forces as 
frictional resistance, gravity, etc., always have in 
some way influenced the motion of the body. 

Newton's second law of motion says that, a 
change in the motion of a body is proportional to 
the force causing the change, and takes place in 
the direction in which the force acts. If several 
forces act on a body, the change is proportional to 



140 SELF-TAUGHT MECHANICAL DRAWING 


the resultant of the several forces, and takes place 
in the direction of the resultant. This has been 
clearly explained in the previous pages, in connec¬ 
tion with the resolution and composition of forces. 
The most important point to note in regard to 
the second law of motion is that when two or 
more forces act on a body at the same time, each 
causes a motion exactly the same as if it acted 
alone; each force produces its effect independently, 
but the total effect on the motion of the body, of 
course, is a combination of all these independent 
motions. 

Newton^s third law says that for every action 
there is an equal reaction. This means that if a 
force or weight presses downward on a support 
with a certain pressure, the reaction, or resistance 
in the support, must equal the same pressure. If 
a bullet is shot from a rifle with a certain force, 
there is a reaction, or “recoil,’’ in the rifle, equal 
to the force required to give the velocity to the 
bullet. This law is very important, and many 
failures in machine design have been due to 
ignorance of the real meaning of the law of action 
and reaction. 

Newton’s third law may be illustrated by a loco¬ 
motive drawing a train of cars. The driving 
wheels give as much of a backward push on the 
rails as there is of forward pull exerted on the 
train; and it is only because the rails are held in 
place by their fastenings, and by the weight rest¬ 
ing on them, that the locomotive is able to pull the 
train forward. This principle of action and reac¬ 
tion being equal and opposite is also an effectual 


ELEMENTS OF MECHANICS 


141 


bar to any perpetual-motion machine, as such a 
machine in order to work would have to produce 
a greater action in one direction than the reaction 
in the other direction. 

The Pendulum.—A body or weight suspended 
from a fixed point by a string or rod, and free to 
oscillate back and forth is called a pendulum. The 
center of oscillation is the point which, if all of the 
material composing the pendulum, including the 
sustaining string or rod, were concentrated at it 
(the material so concentrated being considered as 
being suspended by a line of no weight) would 
vibrate in the same time as the actual pendulum. 
The length of the pendulum is the length from the 
point of suspension to the center of oscillation. 

When the length of the pendulum is unchanged, 
its time of vibration will be the same, if its angle 
of vibration does not exceed three or four degrees, 
and its time of vibration will be but slightly in¬ 
creased for larger angles. 

The time of vibration of a pendulum is not 
affected by the material of which it is made, 
whether light or heavy, except as the light mate¬ 
rial will offer greater resistance to the air, by 
presenting a greater surface in proportion to its 
weight, than a heavy material. 

The time of vibration of a penduluni of a given 
length is inversely as the square root of the inten¬ 
sity of gravity. As the intensity of gravity de¬ 
creases with the distance from the center of the 
earth it follows that a pendulum will vibrate faster 
at the poles or at sea level than it will at the equa¬ 
tor or at an elevation. 


142 SELF-TAUGHT MECHANICAL DRAWING 


The time of vibration of a pendulum varies di¬ 
rectly as the square root of its length. That is, a 
pendulum to vibrate in one-half or one-third the 
time of a given pendulum will need to be only one- 
quarter or one-ninth of its length. 

Example 1 ,—A pendulum in the latitude of New 
York will require to be 39.1017 inches long to beat 
seconds. Required the length of a pendulum to 
make 100 beats per minute. 

A pendulum to make 100 beats per minute will 
have to make its vibrations in 60-100 of the time 
of one which is making 60 beats per minute, and 
its length will be equal to the length of one which 
beats seconds, multiplied by the square of 60-100, 
or: 


39.1017 X 60 _ 39.1017 X 3600 

10010,000 


14.076 inches. 


Example 2 ,—Required the time of vibration of 
a pendulum 120 inches long. Letting x repre¬ 
sent the required time, we have the proportion 

V1^: V393017 = ic: 1 , or 10.954 : 6.253 = a:: 1 . 


X 


10.954 

6.253 


1.75 second. 


A short pendulum may be made to vibrate as 
• slowly as desired by having a second‘‘bob” placed 
above the point of suspension, which will partially 
counteract the weight of the lower bob. 

Falling Bodies.—A falling body will have ac¬ 
quired a velocity at the end of the first second of 
32.16 feet per second, under ordinary conditions. 
If the body is of such shape or material as to pre¬ 
sent a large surface to the air in proportion to its 







ELEMENTS OF MECHANICS 


143 


weight, its velocity will, of course, be lessened, and 
as its velocity depends upon the force of gravity, 
its velocity will be affected somewhat by the lati¬ 
tude of the place, and its distance above sea level. 
During the next second it will acquire 32.16 feet 
additional velocity, giving it a velocity of 64.32 
feet at the end of the second second. Each suc¬ 
ceeding second will add 32.16 feet to the velocity 
the body had at the end of the preceding second. 

To find the velocity of a falling body at the end 
of any number of seconds, therefore, multiply the 
number of seconds during which the body has 
fallen by 32.16. This rule, expressed as a formula, 
would be: 

V = 32.16 X t 

in which v = velocity in feet per second, t = time 
in seconds. 

The acceleration due to gravity, 32.16 feet, is 
often, in formulas, designated by the letter g. As 
an example, find the velocity of a falling body at 
the end of the twelfth second: 

V = 32.16 X 12 = 385.92 feet. 

As the body falling starts from a state of rest, 
its average velocity will be one-half of its final ve¬ 
locity; the distance through which it falls equals 
the average velocity multiplied by the number of 
seconds during which it has been falling. This 
rule, expressed as a formula, is: 

/i = Y X i 

in which h = distance or height through which 



144 SELF-TAUGHT MECHANICAL DRAWING 


body falls, and v and t have the significance given 
above. But -y = 36.16 X if this value of v is in¬ 
serted in the formula just given, we have: 



32.16 X t X t 

2 


16.08 


This last formula, expressed in words, gives us 
the rule that the distance through which a body 
falls in a given time equals the square of the num¬ 
ber of seconds during which the body has fallen, 
multiplied by 16.08. 

How long a distance will a body fall in 10 sec¬ 
onds? Inserting ^ = 10 in the formula, we have: 

h = 16.08 = 16.08 X 10- = 16.08 X 100 = 1608 

feet. 

The time, in seconds, required for a body to fall 
a given distance equals the square root of the 
distance, expressed in feet, divided by 4.01. Ex¬ 
pressed as a formula, this rule would be: 

t = XT. 

4.01 

As an example, assume that a stone falls through 
a distance of 3600 feet. How long time is required 
for this? 

Inserting h = 3600 in the formula, we have: 

V 36(» 60 . _ 

t = - "" 4 Qf "" seconds, very nearly. 

The velocity of a falling body after it has fallen 
through a given distance equals the square root of 
the distance through which it has fallen multi¬ 
plied by 8.02. 






ELEMENTS OF MECHANICS 


145 


This rule, expressed as a formula, is: 

V = 8.02 V h. 

What is the velocity of a falling body after it 
has fallen through a distance of 3600 feet? 

Inserting h = 3600 in the formula, we have: 

'y = 8.02 X V360'0 = 8.02 X 60 = 481.2 feet. 

The height from which a body must fall to acquire 
a given velocity equals the square of the velocity 
divided by 64.32. As a formula, this rule is: 


64.32 


From what height must a body fall to acquire a 
velocity of 500 feet per second? Inserting v = 500 
in the formula given, we have: 


500 ^ 

64.32 


500 X 500 

64.32 


= 3887 feet. 


If a body is thrown upward with a given ve¬ 
locity, its velocity will diminish during each second 
at the same rate as it increases when the body 
falls. A body thrown up into the air in a vertical 
direction will return to the ground with exactly 
the same velocity as that with which it was thrown 
into the air. At any point, the velocity on the up¬ 
ward journey will be equal to the velocity on the 
downward journey, except that the direction is 
reversed. 

The acceleration of a falling body, 32.16 feet per 
second, is the value at the latitude of New York, 
at sea level. 

The force required to give to a falling body its 





14G SELF-TAUGHT MECHANICAL DRAWING 


acceleration of 32.16 feet per second is the weight 
of the body itself. The force required to give any 
acceleration to a body, then, is to the weight of the 
body as that acceleration is to the acceleration 
produced by gravity. Therefore, to find the force 
required to produce a given rate of acceleration to 
a body, divide the weight of the body by 32.16, 
and multiply the quotient by the required rate of 
acceleration. 

Example .—A body weighing 125 pounds is to be 
lifted with an acceleration of 10 feet per second. 
Required the strain on the sustaining rope. 

125 

on i X 10 = 38.8, the tension necessary to produce 
oz. lb 

the acceleration. 

To this must be added the pull necessary to lift 
the weight without acceleration, or the weight of 
the body itself. Thus 38.8 + 125 = 163.8 is the re¬ 
quired tension on the rope. 

The rate of acceleration which a continuously 
acting force will produce is equal to the force 
divided by the weight of the body, multiplied by 
32.16. 

Energy and Work.—The unit of work, the stand¬ 
ard by which work is measured, is foot-pound, 
or the amount of work done in lifting a weight or 
overcoming a resistance of one pound through one 
foot of space. 

‘‘Energy is the product of a force factor and a 
space factor. Energy per unit of time, or rate of 
doing work, is the product of a force factor and a 
velocity factor, since velocity is space per unit of 
time. Either factor may be changed at the ex- 


ELEMENTS OF MECHANICS 


147 


pense of the other; i.e., velocity may be changed, 
if accompanied by such a change of force that the 
energy per unit of time remains constant. Corre¬ 
spondingly force may be changed at the expense of 
velocity, energy per unit of time being constant. 
Example .—A belt transmits 6000 foot-pounds per 
minute to a machine. The belt velocity is 120 feet 
per minute, and the force exerted is 50 pounds. 
Frictional resistance is neglected. A cutting tool 
in the machine does useful work; its velocity is 20 
feet per minute, and the resistance to cutting is 
300 pounds. Then the energy received per minute 
= 120 X 50 = 6000 foot-pounds; and energy deliv¬ 
ered per minute = 20 X 300 = 6000 foot-pounds. 
The energy received therefore equals the energy 
delivered. But the velocity and force factors are 
quite different in the two cases.’’ (Prof. A. W. 
Smith.) 

Force of the Blow of a Steam Hammer or Other 
Falling Weight .—The question, '‘With what force 
does a falling hammer strike?” is often asked. 
This question can, however, not be answered 
directly. The energy of a falling body cannot be 
expressed in pounds, simply, but must be expressed 
in foot-pounds. The energy equals the weight of 
the falling body multiplied by the distance through 
which it falls, or, expressed as a formula: 

E= WXh, 

in which E = energy in foot-pounds, 

W = weight of falling body in pounds, 
h = height from which body falls in feet. 

The energy can also be found by dividing the 


148 SELF-TAUGHT MECHANICAL DRAWING 


weight of the falling body by 64.32 and then mul¬ 
tiplying the quotient by the square of the velocity 
at the end of the distance through which it falls. 
This rule, expressed as a formula, is: 


IP - sy 2 

^ 64.32 ^ 


in which E and W denote the same quantities as 
before, and v = the velocity of the body at the end 
of its fall. 

Both of these formulas give, of course, the same 
results. That the second method gives the same 
result as multiplying the weight by the height 
through which it falls, is evident from the fact, 
stated under the head of ‘ ‘Falling Bodies, ’’ that the 
square of the velocity of a falling body, divided 
by 64.32, gives the height through which it has 
fallen. 

This second method allows of determining the 
energy of any weight or force moving at a given 
velocity, whether its velocity has been acquired by 
falling, or is due to other causes. 

Now assume that we wish to find the force of 
the blow of a 300-pound drop hammer, falling 2 
feet before striking the forging, and compressing 
it 2 inches. 

The energy of the falling hammer when reach¬ 
ing the forging is: 

E = W Xh = 300 X 2 = 600 foot-pounds. 

This energy is used during the act of compress¬ 
ing the forging 2 inches or 0.166 of a foot. Con¬ 
sequently, the average force of the hammer with 


ELEMENTS OF MECHANICS 149 

which it compresses the forging is 600 0.166 + 

the weight of the hammer, or 

Average force of blow = + 300 = 

U.lbb 

3600 + 300 = 3900 pounds. 

The general formula for the force of a blow is: 

F = + W 

a 

in which F = average force of blow in pounds, 

W = weight of hammer in pounds, 
h = height of drop of hammer in feet, 
d = penetration of blow in feet. 

A horse-power, in mechanics, is the power ex¬ 
erted, or work done, in lifting a weight of 33,000 
pounds one foot per minute, or 550 pounds one foot 
per second. The power exerted by a piston driven 
by steam or other medium during one stroke, in 
foot-pounds, is equal to the area of the piston, 
multiplied by the pressure per square inch, multi¬ 
plied by the stroke in feet, the product of the area 
by the pressure giving the force, and the stroke 
giving the distance through which the force is 
exerted. In the case of steam engines, where the 
steam is cut off at one-quarter, one-third or one- 
half of the stroke, the piston being driven the rest 
of the way by the expansion of the steam, the 
average pressure for the entire stroke, the ‘^mean 
effective pressure^ ^(M.E.P.), as it is called, is the 
basis of calculations. As each revolution of the 
engine equals two strokes of the piston, the number 
of foot-pounds per minute an engine is developing 
will be the product of the area of the piston in 




150 SELF-TAUGHT MECHANICAL DRAWING 


square inches, multiplied by the mean effective 
pressure, multiplied by the stroke in feet, multi¬ 
plied by the number of revolutions per minute 
times 2. This product, divided by 33,000, gives 
the mdicafecZ horse-power (I.H.P.) of the engine; 
this name being derived from the fact that the 
mean effective pressure is determined by the use 
of the steam engine indicator. Therefore: 

T TT p Are siX M.E.P. X stroke X rev, per min. X2 

* ’ 33,000 

This formula may be transposed in various ways 
to give other information. For instance, if the 
piston area for a given horse-power is desired, 
then 

* LH.P. X 33,000 

A Y*po --i-- 

M.E.P. X stroke X rev. per min. X 2. 

If the volume of the cylinder is desired, then 

A s, 1 I.H.P. X 33000 

Area X stroke = - •- 

M.E.P. X rev. per mm. X 2. 

If the pressure to produce a given horse-power 
is desired, then 

j[/[ p _ I.H.P. X 33000 _ 

Area X stroke X rev. per min. X 2. 

The mean effective pressure in the cylinder of 
the engine is, of course, considerably less than the 
boiler pressure as shown by the steam gauge. The 
indicated horse-power of an engine does not take 
into account the losses caused by the friction of 
the working parts. The power which the engine 
actually delivers as shown by a brake dynamo¬ 
meter or other contrivance at the flywheel is called 
the brake horse-power. 









CHAPTER IX 


FIRST PRINCIPLES OF STRENGTH OF MATERIALS 

Factor of Safety.—It is obvious that it would be 
unsafe in designing a piece of construction work 
to allow a strain of anywhere near the breaking 
limit of the material it is to be made from. It is, 
therefore, customary in making any calculations 
for the size of the parts to use what is called a 
factor of safety, by making the part from three or 
four to ten or even more times the strength neces¬ 
sary to just resist breaking with a steady load. 
The factor of safety used will depend upon several 
considerations. It will depend, first, upon the na¬ 
ture of the material used. A wrought or drawn 
metal, for instance, will be likely to be more uni¬ 
form in its nature than a cast metal which may 
contain air holes, or which may be more or less 
spongy, or which may be under unequal strains in 
cooling. The matter of strains in a casting due to 
unequal cooling is to a considerable extent a mat¬ 
ter of proper or improper design; still it is not 
possible to entirely avoid them. 

Again the factor of safety to be used will depend 
upon the nature of the work which will be re¬ 
quired of the part. If the part has to simply sus¬ 
tain a steady load it will not need to be as strong 
as though the load was applied and reversed, or 

151 


152 SELF-TAUGHT MECHANICAL DRAWING 


even as strong as though the load was applied and 
released. To illustrate, it is a familiar fact that a 
piece of wire which may be bent a given amount 
without apparent injury, may be broken by repeat¬ 
edly bending it back and forth the same amount 
at one point. And, similarly, in machine parts, 
rupture may be caused not only by a steady load 
which exceeds the carrying strength, but by re¬ 
peated applications of stresses none of which are 
equal to the carrying strength. Rupture may also 
be caused by a succession of shocks or impacts, 
none of which alone would be sufficient to cause it. 
Iron axles, the piston rods of steam hammers and 
other pieces of metal subjected to repeated shocks, 
invariably break after a certain length of service. 

The factor of safety used will therefore vary 
widely with the nature of the work required of the 
part. For a steady or “dead” load. Prof. A. W. 
Smith says: “In exceptional cases where the 
stresses permit of accurate calculation, and the 
material is of proven high grade and positively 
known strength, the factor of safety has been 
given as low a value as IJ; but values of 2 and 3 are 
ordinarily used for iron or steel free from welds; 
while 4 to 5 are as small as should be used for cast 
iron on account of the uncertainty of its composi¬ 
tion, the danger of sponginess of structure, and 
indeterminate shrinkage stresses.” Others would 
make 3 the lowest factor of safety that should be 
used for wrought iron and steel. 

Where the load is variable, but well within the 
elastic limit of the material, that is where the load 
is not so great but so that the part will immedi- 


STRENGTH OF MATERIALS 


153 


ately resume its original shape when the load is 
removed, a factor of safety of 5 or 6 might be 
used. The part will need to be made stronger if 
the load or force acts first in one direction and 
then in the opposite direction, that is, if it acts 
back and forth, than it will need to be if the same 
force is simply applied and then released. Where 
the part is subjected to shock, the factor of safety 
is generally made not less than 10. A factor of 
safety as high as 40 has been used for shafts in 
mill-work which transmit very variable powers. 

In cases where the forces are of such a nature 
that they cannot be determined, then Prof. Smith 
says: ‘'Appeal must be made to the precedent of 
successful practice, or to the judgment of some ex¬ 
perienced man until one^s own judgment becomes 
trustworthy by experience. * * * In proportioning 
machine parts, the designer must always be sure 
that the stress which is the basis of calculation 
or the estimate, is the maximum possible stress; 
otherwise the part will be incorrectly propor¬ 
tioned.^^ And he cites the case of a pulley where 
if the arms were to bo designed only to resist the 
belt tension they would be absurdly small, because 
the stresses resulting from the shrinkage of the 
casting in cooling are often far greater than those 
due to the belt pull. 

In many cases the practical question of feasi¬ 
bility of casting will determine the thickness of 
parts, independent of the question of strength. 
For instance, on small brass work, such as plumb¬ 
ers^ supply, and small valve work, a thickness of 
about 3-32 of an inch is as little as can be relied 


154 SELF-TAUGHT MECHANICAL DRAWING 


on to make a good casting on cored out work; or 
in the case of partitions in such work where the 
metal has to flow in between cores, a thickness of 
about J of an inch is as small as should be used; 
yet such thicknesses may be much greater than 
are required to give the necessary strength. On 
larger cast iron work, the thickness to be allowed 
to insure a good casting will, of course, depend 
upon the size of the piece. The judgment of the 
pattern-maker or foundry-man will naturally de¬ 
termine the thickness in such cases. 

Shape of Machine Parts.—While the size of ma¬ 
chine parts will vary greatly with the nature of the 
work required of them, their shape will depend 
very much on the manner or direction in which 
the load or strain is brought to bear upon them. 
If the part is subjected to simple tension, that is, 
merely resists a force tending to pull it apart, then 
the shape of the member which serves this purpose 
is not very material, though a round rod, being most 
compact and cheapest, is best. Almost any shape 
will answer, however, though it is well to avoid 
using thin and broad parts, as a strain, though not 
greater than that which the part as a whole might 
bear safely, might be brought upon one edge, pro¬ 
ducing a tearing effect beyond the safe limit. For 
resisting simple tension the part should be made of 
uniform size its entire length, of a size to be deter¬ 
mined by the tensile strength of the material and 
the factor of safety used. 

If the part is to resist compression, then when 
the proportion of its length to its diameter or 
thickness is such that it will ‘‘buckle'’ or bend, 


STRENGTH OF MATERIALS 


155 


instead of crushing, that is when its length ex¬ 
ceeds five or six times its diameter, it becomes 
desirable to use a hollow or cross-ribbed form of 
construction, so as to get the metal as far from the 
axis of the piece as possible. The hollow cylind¬ 
rical form, by getting all of the metal equally dis¬ 
tant from the axis is, of course, most effective, 
but considerations of appearance may make a hol¬ 
low square form more desirable, while considera¬ 
tions of cost may make a cross-ribbed form to be 
preferred, as such a form can be cast without the 
use of cores. In cases where a wrought metal must 
be used a solid form is often the only practicable 
one. When it becomes important to keep the 
weight down to the lowest point, it is common to 
have the piece slightly enlarged in the middle 
of its length, as in the case of connecting rods of 
steam engines. In the case of steam engine con¬ 
necting-rods, the tendency to buckle is least side¬ 
ways, as the cross-head and crank-pins tend to 
hold it in line this way, while the rotary motion of 
the crank-pin tends to produce buckling the other 
way. Connecting rods are therefore frequently 
made somewhat flat, of a breadth about twice 
their thickness. 

When a piece is designed to resist bending, it 
becomes desirable to get a good depth of material 
in the direction in which the force is applied, as 
the capacity of a piece to resist bending increases 
as the square of its thickness or depth in the di¬ 
rection of the force, but only directly as its breadth 
or width, so that to increase the thickness of a 
piece two or three times in the direction of the 


156 SELF-TAUGHT MECHANICAL DRAWING 


force would increase its capacity to resist bending 
four or nine times; while to increase its breadth 
two or three times would only increase its strength 
two or three times. The proportion of depth to 
breadth which can be used will, of course, depend 
upon the length of the piece, as if the piece is long 
and its depth is made large in proportion to its 
thickness the tendency will be for the piece to 
buckle, or yield sideways. To resist this tendency 
it is customary to put ribs on the edges of such a 


c—) 


Fig. 130. 



Fig. 131. 


Figs. 130 and 131. — Beam Cross-sections of 
Different Types. 


piece, giving it the form shown in Fig. 130. The 
hollow box-form shown in Fig. 131 is, of course, 
equally effective to resist combined bending and 
buckling stresses, and in some cases may be pref¬ 
erable as a matter of appearance on account of 
the impression of solidity which it gives. 

A projecting beam, like that shown in Fig. 132, 
designed to resist a force or sustain a load at 
its end, would need to have its lower edge made 
of the form of a parabola, if made of uniform 
thickness. If the edges were ribbed to prevent, 
buckling, then material might be taken out of 
the middle portion, as shown in Fig. 133, without 
weakening it. 











STRENGTH OF MATERIALS 


157 


Strength of Materials as Given by Kirkaldy's 
Tests.—A very large number of tests of cast iron 
made by Kirkaldy gave results as follows: Tensile 
strength per square inch, necessary to just tear 
asunder, from about 10,000 or 12,000 pounds to 
about 28,000 or 32,000 pounds, or an average 
strength of about 20,000 pounds. Tests on the 
ability of cast iron to resist crushing gave results 
varying from about 50,000 to about 150,000 pounds. 



Fig. 132. — Cantilever of Fig. 133. — Common Design 


Uniform Strength, when 
Loaded at End. 


of Cantilever of Uniform 
Strength. 


or an average strength of about 100,000 pounds 
per square inch. These tests indicate that cast 
iron has about five times the capacity to resist 
crushing that it has to resist tension. They also 
indicate that cast iron is a somewhat uncertain 
material. 

Tests of wrought iron indicated a tensile strength 
of between 40,000 and 50,000 pounds per square 
inch, the elastic limit being reached at about one- 
half the tensile strength. Tests on steel castings 
gave results for tensile strength ranging from 
55,000 to about 64,000 pounds per square inch. 












158 SELF-TAUGHT MECHANICAL DRAWING 


the elastic limit being reached at about 30,000 
pounds. 

Tests- of wire gave results as follows: Brass, 
from 81,000 to 98,000 pounds per square inch of 
area. Iron, from 59,000 to 97,000 pounds. Steel, 
from 103,000 to 318,000 pounds. 

The tensile strength of regular machine steel 
(low carbon steel) is generally given at about 
60,000 pounds per square inch. 

Size of Parts to Resist Stresses.—To resist ten¬ 
sion it is, of course, only necessary to have the 
piece of such a size that each square inch shall not 
have a stress greater than the average strength 
of the material (as 20,000 pounds for cast iron) 
divided by whatever factor of safety may be 
selected. 

To Resist Crushing.—Prof. Hodgkinson^s rule 
for the strength of hollow cast iron pillars is as 
follows: To ascertain the crushing weight in tons 
multiply the outside diameter by 3.55; from this 
subtract the product of the inside diameter multi¬ 
plied by 3.55, and divide by the length multiplied 
by 1.7. Multiply this quotient by 46.65. Ex¬ 
pressed as a formula this rule would be: 


Sc = 46.65 X 


{D X 3.55) - (d X 3 .55} 
L X 1.7 


in which 

Sc = ultimate compressive (crushing) strength 
of hollow column, in tons, 

D = outside diameter in inches, 
d = inside diameter in inches, 

L = length of column in feet 




STRENGTH OF MATERIALS 


159 


Any desired factor of safety may be introduced 
in the above formula by dividing the factor 46.65 
by the factor of safety. In this case the formula 
would be: 

o _ 46.65 X r(Z) X 3. 55) - (d X 3.55)1 

F X L X 1.7 

in which 

5s = safe compressive strength in tons, 

F = factor of safety, and 

D, d and L have the same meaning as above. 

This rule and formula assumes that the ends of 
the column are perfectly flat and square, and that 
the load bears evenly on the whole surface. 

If the ends are rounded, the column yields at 
about one-half the stress of one with fixed square 
ends. 

To Resist Bending.—In the following commonly 
given rules for the strength of beams or bars to 



Fig. 134.—Rectangular Cantilever. 


resist breaking by transverse stresses, the tensile 
strength of cast iron is assumed at 20,000 pounds 
per square inch. Divide 20,000 in the formulas 



















160 SELF-TAUGHT MECHANICAL DRAWING 

by the desired factor of safety. The breadth and 
depth of rectangular bars, the diameter, if the bar 
is round, and the length, are all in inches. 

For rectangular bars fixed at one end with the 
force applied at the other. Fig. 134, the breaking 
load equals 

_1_ ^ bX 20,000 
6 ^ I 

t 

For round bars under the same conditions. Fig. 
135, the breaking load equals 

1 .. 0.59 X d^X 20,000 

' T - 1 - 

If the rectangular bar is hollow, as shown in 



Fig. 135.—Circular Section Cantilever. 


Fig. 131, subtract the internal b X from.the 

external bX d~. 

If the round bar is hollow subtract the internal 
d ^ from the external d ^ 

The case of a bar of the I-section shown in Fig. 
130 is similar to that of the hollow rectangular bar 
of Fig. 131, the depressions in its sides correspond¬ 
ing to the hollow part of Fig. 131, the sum of their 














STRENGTH OF MATERIALS 161 

depths corresponding with the internal width b of 
the hollow rectangular bar. 

If a beam is fixed at one end and the load is 
evenly distributed throughout its entire length, 
it will bear double the weight it will if the load 
is supported at the outer end. 

If the beam is supported at the ends and loaded 
in the middle it will bear four times the weight of 
the beam of Fig. 134, or, if the load is evenly dis¬ 
tributed throughout the length of the beam, eight 
times. 

If the beam, instead of being simply supported 
at the ends, has the ends fixed and is loaded at the 
center, its ability to resist breaking will be doubled 
as compared with that when loaded at the center 
and with the ends only supported. 

Regarding the safe load that beams or bars of 
different material may bear Grifhn says that'‘with 
but a general knowledge of the elastic limit, ordi¬ 
nary steel is good for from between 12,000 to 15,000 
pounds per square inch non-reversing stress, and 
from 8000 to 10,000 pounds reversing stress. Cast 
iron is such an uncertain metal, on account of its 
variable structure, that stresses are always kept 
low, say from 3000 to 4000 for non-reversing stress, 
and 1500 to 2500 for reversing stress.'^ 

Again, though the tests of wrought iron show it 
to have a much higher tensile strength than cast 
iron, Nystrom, in formulas for lateral strength, 
gives wrought iron but little more than three-quar¬ 
ters the value of cast iron, probably because it 
bends so readily. 


162 SELF-TAUGHT MECHANICAL DRAWING 


A table is appended giving the average breaking 
strength, in pounds per square inch, of some com¬ 
monly used materials in engineering practice. 



Tension. 

Compression. 

Aluminum. 

15,000 

12,000 

Brass, cast. 

24,000 

30,000 

Copper, cast. 

24,000 

40,000 

Iron, cast. 

15,000 

80,000 

Iron, wrought. 

48,000 

46,000 

Steel castings. 

70,000 

70,000 

Structural steel .... 

60,000 

60,000 


Stresses in Castings.—Reference has been pre¬ 
viously made to stresses in castings, due to shrink¬ 
age in cooling. If all parts of a casting could be 
made to cool equally fast there would not be much 
trouble in this respect, but as different parts of a 
casting vary in thickness, the time they require 
to cool will yary, and the thick parts remaining 
fluid the longest, will, on cooling, cause a strain on 
the already cool thin parts. In the case of a pulley, 
where the rim and arms are much lighter than the 
hub, the hub on cooling will tend to draw the arms 
to itself and away from the rim, and if the differ¬ 
ence in thickness is great, they may be even found 
to be pulled away so as to show a crack where they 
join the rim. The remedy in such a case would, of 
course, be first, to take out as much of the metal 
from the center of the hub as possible by means 
of a core, and second, to keep the outside of the 
hub as small as would be consistent with strength, 
getting necessary thickness for set screws by hav¬ 
ing a raised place or boss at that point. 















STRENGTH OF MATERIALS 


163 


As these strains are primarily due to unequal 
cooling, it is evident that in order to reduce them 
to the lowest point the first thing to do is to make 
the different parts of the casting of as nearly uni¬ 
form thickness as possible. Where different parts 
of the casting vary in thickness, the change from 
one thickness to the other should be made as grad¬ 
ual as possible. Sharp internal corners should also 
be avoided, as such places are very liable to be 
spongy; the sand from the sharp corner in the 
mould is also very liable to wash away when the 
metal is poured in, and lodge in some other place, 
causing a defective casting. A good “fillet,” as an 
internal round corner is called, which the pattern¬ 
maker may put into the pattern with wax, putty or 
leather, will not be very expensive, and will save 
much trouble in the casting. 

Besides possessing a knowledge of factors of 
safety, proportioning parts to resist various 
stresses and the like, a general knowledge of the 
principles of foundry and machine shop practice is 
essential to properly design machine work. If one 
does not understand foundry work, he will be con¬ 
stantly designing castings which it will be im¬ 
practicable to mould; if not actually impossible of 
moulding, they will be needlessly expensive. And 
in like manner, unless he understands the general 
principles of machine shop practice, his work will 
be giving trouble at that end of the line. 


CHAPTER X 


CAMS 

General Principles.—In designing machinery it 
is frequently desirable to give to some part of the 
mechanism an irregular motion. This is often 
done by the use of cams, which are made of such 
form that when they receive motion, either rotary 
or reciprocating, they impart to a follower the 
desired irregular motion. 

The follower is sometimes flat, and sometimes 
round. When the follower is round it is usually 
made in the form of a wheel or roller, so as to les¬ 
sen the wear and the friction. The follower may 
work upon the edge of the cam, or if round, it 
may work in a groove formed either on the face 
or on the side of the cam. 

The working surfaces of cams with round fol¬ 
lowers are laid out from a pitch line,' so called, 
which passes through the center of the follower. 
The shape of this pitch line determines the work 
which the cam will do. The working surface of 
the cam is at a distance from the follower equal to 
one-half the diameter of the follower. This prin¬ 
ciple of a pitch line holds good whether the cam 
works only upon its edge like the one shown in 
Fig. 139, or whether it has an outer portion to 
insure the positive return of the follower. This 

164 


CAMS 165 

outer portion is frequently made in the form of a 
rim of uniform thickness around the groove. 

Design a Cam Having a Straight Follower Which 
Moves Toward or From the Axis of the Cam, as 
Shown in Fig. 136.—Let it be required that the 
follower shall advance at a uniform rate from a to 



Fig. 136.—Cam with Straight Follower having Uniform 

Motion. 


h as the cam makes a half revolution, this advance 
being preceded and followed by a period of rest of 
a twelfth of a revolution of the cam. 

Divide that half of the cam during the revolu¬ 
tion of which the follower is to be raised from a to 
h, in this case the half at the right of the vertical 
center line, into a number of equal angles, and 















166 SELF-TAUGHT MECHANICAL DRAWING 


divide the distance from a to 6 into the same num¬ 
ber of equal spaces. Mark off the points so ob¬ 
tained onto the successive radial lines as indicated 
by the dotted lines, and at the points where these 
dotted lines intersect the radial lines draw lines at 
right angles to the radial lines to represent the 
position of the follower when these radial lines 
become vertical as the cam revolves. 

A period of rest in a cam is represented by a cir¬ 
cular portion, having the axis of the cam as its 
center. In order, therefore, to obtain the required 
periods of rest, the distances of a and h from the 
center are marked off upon the radial lines c and 
d, these lines being made a twelfth of a revolution 
from the vertical center line, and lines represent¬ 
ing the follower are drawn at these points as be¬ 
fore. To get the return of the follower the space 
from c to cZ is divided into a number of equal 
angles, and the distance from e to/is divided off 
to represent the desired rate of return of the 
follower. In this case the rate of return is made 
uniform, so the distance ef is spaced off equally. 
The distance of these points from the axis is marked 
off upon the radial lines between c and d, and lines 
representing the follower are drawn. 

A curved line, which may be made with the 
aid of the irregular curves, which is tangent to all 
of the lines representing the follower, gives the 
shape of the cam. 

Fig. 137 shows a cam having the conditions as to 
the rise, rest and return of the follower the same 
as the one shown in Fig. 136, the follower, how¬ 
ever, being pivoted at one end. 


CAMS 


167 


Draw the arc ah representing the path of a point 
in the follower at the vertical center line, and 
divide that part of the arc through which the fol¬ 
lower rises into the same number of equal spaces 
as the half circle at the right of the vertical cen¬ 
ter line is divided into angles. Through these 



Fig. 137.—Cam with Pivoted Follower. 


points draw lines, as shown, representing consecu¬ 
tive positions of the working face of the follower. 
The various distances of the follower from the axis 
of the cam are now marked off upon the corre¬ 
sponding radial lines as before. Lines to represent 
the follower are now drawn across each of these 
radial lines, at the same angle to them that the 
follower makes with the vertical center line when 

















168 SELF-TAUGHT MECHANICAL DRAWING 


at that part of its stroke corresponding to the par¬ 
ticular radial line across which the line represent¬ 
ing the follower is being drawn. A curved line 
passing along tangent to all of these lines gives 
the shape of the cam as before. 

Design a Cam with a Round Follower Rising Ver¬ 
tically.—In Fig. 138 the follower has the same uni¬ 
form rise, and the same periods of rest as before. 



A cam with a round follower is less limited in its 
capabilities than one with a straight follower; in 
the one here shown the follower on its return 
drops below the position in which it is shown. 
That part of the cam during which the conditions 
are the same as in the others is divided off and 











CAMS 


169 


the position of the center of the follower upon the 
radial lines is obtained in the same manner as 
before. That part of the cam representing the 
return of the follower is divided into such angles 
as desired, and the distance through which the fol¬ 
lower is to drop as the cam revolves through each 
of these angles is marked off upon the proper 
radial line. A curved line which is now made to 
pass through all of the points so obtained gives 
the pitch line of the cam. 

In drawing such a cam it is not always neces¬ 
sary to fully draw the working faces. The pitch 
line and the method of obtaining it being shown, 
a number of circles representing consecutive posi¬ 
tions of the follower may be drawn. This will 
usually be sufficient. The side view of the cam, 
which in a case like this would naturally be made 
in section, will give opportunity to show any fur¬ 
ther detail that may be desired. 

Design a Cam with a Round Follower Mounted on 
a Swinging Arm.—Fig. 139 shows such a cam, all 
of the conditions as to rise, rest and return of the 
follower being the same as in the cam shown in 
Fig. 138. The cam is divided into the same angles 
as before, and the position of the follower is laid 
out on these radial lines as though it moved ver¬ 
tically. These positions are then modified in the 
following manner: Draw the arc ah representing 
the path of the center of the follower as it rises, 
and extend the dotted circular lines, which repre¬ 
sent successive heights of the follower, from the 
vertical center line to this arc. The distance of 
each of the intersections of the dotted circular 


170 ^^ELt’-TAUGHT MECHANICAL DRAWING 


lines with the arc ah, from the vertical center line 
is then taken with the compasses and is marked 
off upon the same dotted line from the radial line 
at which it terminates, or, where the follower has 
a period of rest, from both of the radial lines 



Fig. 139.—Cam with Roller Follower Mounted on 

Swinging Arm. 


where the period of rest takes place. Thus the dis¬ 
tance of the point 1 from the vertical center line is 
marked back upon the dotted circular line from the 
radial lines m and oi. Point 2 is marked back from 
the radial line o. Point 3 is marked back from the 
line p. By this means the position which the fol¬ 
lower will occupy, when each of the radial lines 
has become vertical, as the cam revolves, is deter- 

















CAMS 


171 


mined. A curved line 'which is made to pass 
through all of these points will be the required 
pitch line of the cam. The method of getting the 
working face of the cam is indicated by the small 
dotted circular arcs, which are drawn with a radius 
equal to that of the follower. It will be noticed 
that, as the follower, on its return, drops below 
the position in which it is shown, it passes to the 
other side of the vertical center line, so that in 
marking off its position from the radial lines x and 
y this must be borne in mind. The question as to 



Fig. 140.—Reciprocating Motion Cam. 


on which side of a radial line the new position of 
the follower will be, may be readily determined by 
imagining the cam to revolve so as to bring that 
particular line vertical. 

Reciprocating Cams.—Fig. 140 shows a straight 
cam, which by a reciprocating motion imparts a 
sideways motion to its follower. The pitch line 
of such a cam may be determined by intersecting 
lines at right angles to each other. As here shown 
the distance through which the follower is to be 
raised is divided into a number of equal spaces by 
horizontal lines, and the distance through which it 
is desired to have the cam move in order to raise 
the follower from one horizontal line to the next 
one is indicated by vertical lines. A curved line 


















172 SELF-TAUGHT MECHANICAL DRAWING 


which is made to pass through the intersections of 
these lines will be the required pitch line of the 
cam. 

If the follower, instead of rising vertically, rose 
at an angle, or if it were mounted on a swinging 
arm, the pitch line would be modified in the same 
manner as that of the cam shown in Fig. 139. 

Cams With a Grooved Edge.—It is sometimes de¬ 
sired to have a revolving cam impart a sideways 



Fig. 141.—Cam with Grooved Edge. 


motion to a follower. This is done by having a 
groove in the edge of the cam, as shown in Fig. 
141. Such a cam may be considered as a modified 
form of a reciprocating cam, and its pitch line may 
be determined in the same way. 

By laying out a development of the pitch lino, or 
of that part of it which is to operate the follower, 
as shown in Fig. 142, horizontal lines, that is, lines 
parallel with the pitch line, may be drawn to indi¬ 
cate successive stages in the movement of the fol¬ 
lower, and lines at right angles to these to indicate 
















CAMS 


173 


the desired movement of the cam. The pitch line 
is then drawn through the intersections of these 
lines as before. 

A Double Cam Providing Positive Return.—In a 
cam like that shown in Fig. 138, where the return 





^ 






Fig. 142.—Development of Cam Action of Grooved-Edge 

Cam in Fig. 141. 

of the follower is insured by a groove in the face 
of the cam, the groove must be slightly broader 
than the diameter of the cam roller to insure free¬ 
dom of action, as, when the cam is forcing the rol- 



Fig. 143.—Double Cam Providing Positive Return. 


ler away from the center, the roller will revolve in 
the opposite direction to that in which it revolves 
when the other face of the cam groove acts on it 
to draw it toward the center, so that unless clear- 
























174 SELF-TAUGHT MECHANICAL DRAWING 

ance is provided, there will be a grinding action 
between the roller and the faces of the cam groove. 
This clearance, however, causes the cam to give a 
knock or blow on the roller each time its action is 
reversed, and the reversal of the direction of the 
revolution of the roller itself causes a temporary 
grinding action. These actions may become ob- 



Fig. 144.—Positive Return Cam with Rollers Mounted on 

Swinging Arms. 

jectionable, especially at high speeds. A method 
which overcomes these objections, and which is 
preferred by some for such work, is shown in Fig. 
143, where the return is secured by a secondary 
cam mounted on the same shaft as the primary 
cam, but acting on a roller of its own. In this case 
there is no reversal of the direction of the revolu¬ 
tion of the rollers, so that the necessity of provid- 












CAMS 


175 


ing clearance does not exist. Where the forward 
and backward motion of the rollers is in a straight 
line passing through the center of the cam shaft, 
as in this case, it is only necessary in designing 
the secondary cam to preserve the distance be¬ 
tween its pitch line and the pitch line of the prim¬ 
ary cam constant, measuring through the center of 
the cam shaft, as shown at x and y. 

If, however, the rollers are mounted on swing¬ 
ing arms, as shown in Fig. 144, so that their for¬ 
ward and backward motion is not in such a straight 
line, then the shape of the secondary cam will be 
subject to modification on principles previously 
explained. It is obviously necessary where this 
method of operation is used, that'provision be made 
to absolutely prevent any change in the relative 
position of the two cams, as by bolting them to¬ 
gether, or, better still, by having them cast 
together in one piece. 

Cams for High Velocities.—In machinery work¬ 
ing at a high rate of speed, it becomes very im¬ 
portant that cams are so constructed that sudden 
shocks are avoided when the direction of motion 
of the follower is reversed. While at first thought 
it would seem as if the uniform motion cam would 
be the one best suited to conditions of this kind, a 
little consideration will show that a cam best suited 
for high speeds is one where the speed at first is 
slow, then accelerated at a uniform rate until the 
maximum speed is reached, and then again uni¬ 
formly retarded until the rate of motion of the 
follower is zero or nearly zero, when the reversal 
takes place. A cam constructed along these lines 


176 SELF-TAUGHT MECHANICAL DRAWING 



Fig. 145,—Uniformly Accelerated Motion Cam. 


















CAMS 


177 


is called a uniformly accelerated motion cam. The 
distances which the follower passes through during 
equal periods of time increase uniformly, so that, 
if, for instance, the follower moves a distance equal 
to 1 length unit during the first second, and 3 
during the second, it will move 5 length units 
during the third second, 7 during the fourth, and 
so forth. When the motion is retarded, it will 
move 7, 5, 3 and 1 length units during successive 
seconds, until its motion becomes zero at the re¬ 
versal of the direction of motion of the follower. 

In Fig. 145 is shown a uniformly accelerated 
motion plate cam. Only one-half of the cam has 
been shown complete, the other half being an exact 
duplicate of the half shown, and constructed in the 
same manner. The motion of the follower is back 
and forth from A to G, the rise of the cam being 
180 degrees, or one-half of a complete revolution. 
To construct this cam, divide the half-circle, AKL, 
in six equal angles, and draw radii HBi , HCi , 
etc. Then divide AG first in two equal parts AB 
and BG, and then each of these parts in three 
divisions, the length of which are to each other as 
1:3:5, as shown. Then with 77 as a center draw 
circular arcs from B, C, B, etc., to, Gi, , etc. 

The points of intersection between the circles and 
" the radii are points on the cam surface. 

If the half-circle AKL had been divided into 8 
equal parts, instead of 6, then the line AG would 
have been divided into 8 parts, in the proportions 
1:3:5:7:7:5:3:1, each division being the same 
amount in excess of the previous division while 
the motion is accelerated, and the same amount 


178 SELF-TAUGHT MECHANICAL DRAWING 


less than the previous division while the motion is 
being retarded. With a cam constructed on this 
principle the follower starts at A from a velocity 
of zero; it reaches its maximum velocity at D; and 
at G the velocity is again zero, just at the moment 
when the motion is reversed. 

A graphical illustration of the shape of the uni¬ 
formly accelerated motion curve is given in Fig. 



Fig. 146.—Development and Projection of Uniformly 
Accelerated Motion Cam Curve. 


146. To the right is shown the development of 
the curve as scribed on the surface of a cylindrical 
cam. This development is necessary for finding 
the projection on the cylindrical surface, as shown 
at the left. To construct the curve, divide first 
the base circle of the cylinder in a number of equal 





























CAMS 


179 


parts, say 12; set off these parts along line AL, as 
shown; only one division more than one-half of the 
development has been shown, as the other half is the 
same as the first half, except that the curve to be 
constructed here is falling instead of rising. Now 
divide line AK in the same number of divisions as 
the half-circle, the divisions being in the proportion 
1:3 : 5 : 5 : 3 :1, Draw horizontal lines from the 
divisions on AK and vertical lines from B, C, D, 
etc. The intersections between the two sets of 
lines are points on the developed cam curve. These 
points are transferred to the cylindrical surface at 
the left simply by being projected in the usual 
manner. 

In order to show the difference between the uni¬ 
form motion cam curve, and that illustrating the 
uniformly accelerated motion, a uniform motion 
cylinder cam has been laid out in Fig. 147. The 
base circle is here divided in the same number of 
equal parts as the base circle in Fig. 146. The 
divisions are set off on line AL in the same way. 
The line AK, however, is divided into a number of 
equal parts, the number of its divisions being the 
same as the number of divisions in the half-circle. • 
By drawing horizontal lines through the division 
points on AK, and vertical lines through points B, 
C, D, etc., points on the uniform motion cam curve 
are found. It will be seen that this curve is merely 
a straight line AM. The curve is transferred to its 
projection on the cylinder surface at the left, as 
shown. 

It is evident from the developments of the two 
curves in Figs. 146 and 147, that the uniform motion 


180 SELF-TAUGHT MECHANICAL DRAWING 

curve, Fig. 147, causes the follower to start very 
abruptly, and to reverse from full speed in one 
direction to full speed in the opposite direction. 
The uniformly accelerated motion curve. Fig. 146, 
permits the follower to start and reverse very 
smoothly, as is clearly shown by the graphical 



Fig. 147. — Development and Projection of Uniform Motion 

Cam Curve. 


illustration of the curve. The abrupt starting and 
reversal of the follower in the uniform motion 
curve is the cause why this form of cam, while 
the simplest of all cams to lay out and cut, cannot 
be used where the speed is considerable, without 
a perceptible shock at both the beginning and the 
end of the stroke, 






























CAMS 


181 


Besides the uniformly accelerated motion cam 
curve, quite commonly called the gravity curve, 
on account of it being based on the same law of 
acceleration as that due to gravity, there is another 
curve, the harmonic or crank curve, which is quite 
often used in cam construction. The harmonic 
motion curve provides for a gradual increase of 
speed at the beginning, and decrease of speed at the 
end, of the stroke, and in this respect resembles 



Fig. 148.—Lay-out of Harmonic Motion Cam Curve. 


the uniformly accelerated motion curve; but the 
acceleration, not being uniform, does not produce so 
easy working a cam as the gravity curve provides 
for. The harmonic motion curve is, however, very^ 
simple to lay out, and for ordinary purposes, where 
excessively high speeds are not required of the 
mechanism, cams laid out according to this curve 
are very satisfactory. 

The harmonic curve is laid out as shown in Fig. 
148. Draw first a half-circle AEL Divide the 





















182 SELF-TAUGHT MECHANICAL DRAWING 

circle in a certain number of equal parts. Draw a 
line /i, and divide this line in a number of equal 
parts, the number of divisions of being the 
same as that of the half-circle. Now draw hori¬ 
zontal lines from the divisions A, B, C, etc., on the 
half-circle, and vertical lines from the divisions on 
line /i. The points where the lines from corre¬ 
sponding division points intersect, are points on 
the required harmonic cam curve. 

An approximation of the uniformly accelerated 
motion or gravity curve can be drawn as shown in 



Fig. 149.— Approximation of Uniformly Accelerated Motion 

Curve. 

Fig. 149. By using this approximate method, any 
degree of accuracy can be attained without the 
necessity of dividing the vertical line AK, Fig. 
146, in an excessively great number of parts. The 
approximate curve in Fig. 149 is constructed as 
follows: Draw a half-ellipse AEI, in which the 
minor axis is to the major axis as 8 to 11. Divide 
this half-ellipse in any number of equal parts, and 
divide the line AiA in the same number of equal 
parts. Now draw horizontal lines from the division 






















CAMS 


183 


points on the ellipse, and vertical lines from^i, 
Cl, etc. The points of intersection between 
corresponding horizontal and vertical lines, are 
points on the cam curve. This cam curve, as well 
as the one in Fig. 148, can be transferred to the 
cylindrical surface of a cylinder cam by ordinary 
projection methods, as shown in Figs. 146 and 147. 

In Figs. 150 and 151 are shown two plate cams 
for comparison. The one in Fig. 150 is a uniform 



Fig. 150.—Plate Cam Laid Fig. 151.—Plate Cam Laid 
out for Uniform Motion. out for Uniformly Accel¬ 

erated Motion. 


motion cam. The dwell is 180 degrees, the rise, 90 
degrees, and the fall, 90 degrees. As shown by 
the sudden change of direction of the cam curve 
at A and B, there is considerable shock when the 
follower passes from its‘‘dwell’Ho the “rise,” as 
well as at the end of the ‘ ‘ fall. ’ ’ A sudden reversal 
takes place at C, which also causes a shock in 
the mechanism connected with the follower. In the* 
uniformly accelerated motion cam. Fig. 151, the 











184 SELF-TAUGHT MECHANICAL DRAWING 

passing from ‘'dwell” to “rise,” the reversal of the 
direction of motion, and the return to the “dwell” 
position, is accomplished by means of smoothly 
acting curves, and, even at high speeds, no per¬ 
ceptible shock will be noticed. 

The examples given will show the necessity of 
careful analysis of conditions, before a certain type 
of cam curve is selected. In machinery which 
works at a low rate of speed, it is not important' 
whether the follower moves with a uniform, har¬ 
monic, or uniformly accelerated motion; but when 
the cam has a high rotative speed, and the follower 
a reciprocating motion, it often becomes practically 
impossible to make use of the uniform motion 
curve in the cam. In such cases, as already men¬ 
tioned, the harmonic, or, preferably, the uniformly 
accelerated motion curve should be used in laying 
out the cam. 


CHAPTER XI 


SPROCKET WHEELS 

When it is desired to transmit power from one 
shaft to another one quite near to it, especially if 
the power to be transmitted is considerable, so as 
to preclude the use of belting, sprocket wheels 
with chain are frequently used, if the speed is not 
high. Bicycles afford a familiar illustration of 
this sort of power transmission. 

Fig. 152 shows a sprocket wheel of a type similar 
to those used on bicycles and shows the method of 
getting the shape of the teeth. The chain is shown 
with the links (on the side toward the observer) 
removed so as to allow of showing the teeth with¬ 
out dotted lines. The size of a sprocket wheel to 
fit a given chain may be determined graphically as 
follows: A circle, not shown in the illustration, is 
first drawn of a diameter about equal to that of 
the desired wheel, and this circle is spaced off into 
as many divisions as the wheel is to have teeth. 
Lines corresponding to the dotted radial lines in 
the upper half of the wheel shown, are drawn from 
these division points to the center of the circle. A 
templet, similar in shape to that shown in Fig. 154, 
is next cut out of paper, the lines ah and cd being 
at right angles to each other, and the length of a 
link of the chain, measured from center to center 

185 


186 SELF-TAUGHT MECHANICAL DRAWING 

of the pins as shown at a, Fig. 152, is marked off ^ 
upon the line ah, measuring equally each way from 
the center line cd. In getting the length of the 
link in the chain it will be best, for the sake of ac¬ 
curacy, to measure off the length of a considerable 
portion of the chain, and with the spacing com¬ 
passes divide this length into twice as many spaces 
as there are links in the measured portion of the 



Fig. 152.—Sprocket Wheel and Chain. 


chain. The compasses, being then set to exactly 
half the length of a link, may be used to mark off 
the length of the link, 1 — 2, upon the templet. 
Now letting the angle abc, Fig. 155, represent one 
of the angles into which the circle has been di¬ 
vided, bisect it to get a center line bd, and placing 
the templet so that its line cd shall coincide with 
this center line move it along until the points 1—2 
shall coincide with the lines ab and cb of the angle. 
These points being now marked off upon the lines, 
give the location of the centers of the pins in the 
chain, and a line connecting them will be one side 































SPROCKET WHEELS 


187 


of the polygon which forms the pitch line of the 
wheel. A spiral may now be formed upon this 
polygon (see geometrical problem 19, Figs. 41 and 
42), and will give the path of the pin as the chain 


Fig. 153.— Sprocket Wheel Designed for Common 

Link Chain. 

unwinds from the wheel when the latter revolves, 
as shown in Fig. 152. The working face of that 
part of the tooth in the wheel lying outside of the 
pitch polygon is now struck from such a center as 
will cause it to fall slightly within the path of the 
chain, as just obtained, so that the link may fall 


c 



Fig. 154. Fig. 155. 

Figs. 154 and 155.—Graphical Method of Laying Out 

Sprocket Wheel. 

freely into place as it enters upon the tooth. Of 
course allowance must be made all around for the 
natural roughness of the casting if the wheel is to 
be left unfinished. The length of the tooth is 
usually made about equal to the width of the chain. 

























188 SELF-TAUGHT MECHANICAL DRAWING 


If a wheel is to have many teeth, it will gener¬ 
ally be accurate enough to consider the pitch line 
as a circle of a circumference equal to the number 
of the teeth multiplied by the length of the link. 
Its diameter will then, of course, be found by 
dividing the circumference by 3.1416. 

In the case of the wheel shown in Fig. 152, 
should the pitch line be regarded as such a circle 
it would have a diameter a little over a thirty- 
second of an inch too small, if the length of the 
link is taken at three-quarters of an inch. If the 
wheel were to be made twice as large, the error 
would be a little/css than a sixty-fourth of an inch, 
as it would decrease at a slightly faster rate than 
that at which the number of the teeth increased. An 
error of a sixty-fourth of an inch in the diameter 
of such a sprocket would be of but very little 
moment. Where a sprocket has but few teeth, 
however, it will be on the side of safety to always 
give to the pitch line its true polygonal form, and 
the only way by which its diameter could be ascer¬ 
tained with any greater accuracy than by the 
method here given would be to calculate it, as may 
be done by trigonometry. When the pitch line of 
a sprocket is regarded as a circle, the path of the 
chain as it unwinds will be regarded as an involute 
(see geometrical problem'20). 

The shape of the rim of a sprocket wheel will be 
governed by the style of the chain for which it is 
designed. Fig. 153 shows a portioi) of the rim of 
a wheel which is designed for a common link 
chain; but whatever the general shape of the rim 
may be, the working faces of the teeth, or of the 


SPROCKET WHEELS 


189 


projections which correspond to teeth, will always 
be made on the principles here explained. 

The speed ratio of the two wheels of a pair of 
sprockets will be inversely as the number of teeth 
in each. For instance, if the large and the small 
wheels have respectively 13 and 7 teeth, then the 
speed of the large wheel will be to the speed of 
the small wheel as 7 to 13. 


CHAPTER XII 

GENERAL PRINCIPLES OF GEARING 

Friction and Knuckle Gearing.—In machinery 
it is frequently necessary to transmit power from 
one shaft to another near to it. For this purpose 
gears are generally employed. Let a and 6, Fig. 
156, be two such shafts. If now disks c and d are 
mounted upon these shafts, of such diameters as 



to give the required speed ratio, we will have 
gearing in its simplest form. Such disks, having 
their edges covered with leather or other equiva¬ 
lent material, are called friction gears and are 
sometimes employed on light work. At best, how¬ 
ever, they will transmit but little power. 

If now we make semi-circular projections at 
equal distances apart upon the outside of the cir¬ 
cles c and d, and cut out corresponding depressions 
inside of the circles, as shown in Fig. 157, we will 
have a simple form of toothed gearing and the cir- 

190 









GENERAL PRINCIPLES OF GEARING 


191 


cles c and d will be the pitch circles. Such gears, 
called knuckle gears, are sometimes employed on 
slow-moving work where no special accuracy is 
required. They will not transmit speed uniformly. 
If the driver of such a pair of gears rotated at a 
uniform rate, the driven gear would have a more 
or less jerky movement as the successive teeth 
came into contact, and if run at high speed they 
would be noisy. Various curves may be employed 
to give to gear teeth such an outline that the 
driver of a pair of gears will impart a uniform 
speed to the driven one, but in common practice 
only two kinds are used, the cycloidal, or, as it is 
sometimes called, epicycloidal, and the involute. 

Epicycloidal Gearing.—Let the circles a, b and 
c, Fig. 158, having their centers on the same 
straight line, be made to rotate so that their cir¬ 
cumferences roll upon each other without slipping. 
If the circle c has tracing points 1, 2, 3 upon its 
circumference, and when we start to rotate the 
circles point 1 is half way around from the posi¬ 
tion in which it is shown, then in rotating the cir¬ 
cles sufficiently to bring the tracing points to the 
position in which they are shown, point 1 will 
trace the line 1' inwardly from the circle a, and the 
line 1 " outwardly from the circle h. Point 2 will 
trace the two lines which are shown meeting at 
that point, one inwardly from the circle a, and one 
outwardly from the circle h. Point 3 will similarly 
trace the two lines which met at that point. Inas¬ 
much as these lines were traced simultaneously by 
points at a fixed distance apart, it is evident that 
if the circle c were to be removed, and the circles 


192 SELF-TAUGHT MECHANICAL DRAWING 

a and b were rolled back upon each other, these 
lines would work smoothly together, being in con¬ 
tact and tangent to each other at all times upon 
the line of the circles. If the circle cis now placed 
beneath the circle b in the position shown, and the 
three circles are rolled together as before, the tra¬ 
cing points would trace lines inwardly from 6, and 



cycloidal Gearing, lute Gearing. 


outwardly from a, which would also work together 
smoothly if the circle c were removed and the cir¬ 
cles a and b were rolled back upon each other. It 
is evident that as the three circles are rolled 
together the lines formed by the tracing points are 
the same as though either a or b were taken by 
itself, and the circle c were rolled either within 
or upon it, hence the lines formed by the tracing 
points are either epicycloids or hypocycloids as 
the case may be, and so could be formed by the 








GENERAL PRINCIPLES OF GEARING 193 

plotting method described in the geometrical 
problems. 

If these two sets of lines are now joined together 
so that the lines which extend inwardly from a or 
b form a continuation of those which extend out¬ 
wardly and reverse curves are made at a distance 
from the first set equal to the thickness of a gear 
tooth, and they are then cut off at such a distance 
both outside and inside of the circles a and b as to 
give to the teeth the proper 
length, it is evident that 
we will have a pair of per¬ 
fectly working gears. The 
circles a and b would roll 
upon each other without 
slipping and hence would 
be true pitch circles. The 
teeth would work smoothly 
together in constant contact, the point of contact 
being always on the line of the generating circle. 

The length of the point of the gear tooth, that 
is the portion lying outside of the pitch line, is 
usually made one-third of the circular pitch—the 
latter being the distance between the teeth meas¬ 
ured from center to center on the pitch line. The 
distance below the pitch line is made somewhat 
greater for the sake of clearance. For the names 
of the various parts of a gear tooth see Fig. 160. 
Cast gears have some backlash between the teeth 
to allow for the roughness of the castings, as 
shown in Figs. 161 and 163. 

It is evident that if another circle, either larger 
or smaller, were substituted for b in Fig. 158, the 



CLEARANCE 


Fig. 160. — Definitions of 
Gear Tooth Terms. 





194 SELF-TAUGHT MECHANICAL DRAWING 


lines formed by the generating circle c either 
within or upon the circle a would remain unchanged. 
Or if a different circle were substituted for a, the 
curves formed within or upon b would remain un¬ 
changed. Hence it follows that all gears in the 
epicycloidal system, having their teeth formed by 
the same generating circle and made of the same 




Fig. 162.—Rack with 
Epicycloidal Teeth. 


Fig. 161.—Gears with Epicycloidal 
Teeth. 

size, will work together correctly, or, as it is com¬ 
monly expressed, are interchangeable. 

In standard interchangeable gears the generat¬ 
ing circle is made one-half the diameter of the 
smallest gear of the set, which has twelve teeth. 
This smallest gear will have radial flanks, as that 
part of the working surface lying within the pitch 
line is called, because the hypocycloid of a circle 
formed by a generating circle of half its size will 
be a straight line passing through its center. 

Fig. 161 shows a portion of a pair of such gears. 
Fig. 162 showing the rack. 










GENERAL PRINCIPLES OF GEARING 


195 


Gears with Strengthened Flanks.—A further ex¬ 
amination of Fig. 158 will show that the curves 
formed by the generating circle when it is in the 
upper of the two positions in which it appears, 
work together by themselves, and those formed 
when it is in the lower position work similarly, so 
that it is not necessary that the same sized gener- 



Fig. 163.—Gears with Involute Teeth. 



Fig. 164.—Rack with 
Involute Teeth. 


ating circle should be used in both positions, unless 
the gears are to be members of an interchangeable 
set of gears. Advantage may be taken of this fact 
to strengthen the roots of the teeth in a pinion. 

If, for instance, in Fig. 161, a smaller generating 
circle were used in the upper position, the effect 
would be to broaden out the roots of the teeth in 
the pinion, and to correspondingly round off the 
points of the teeth of the other gear. 

Gears with Radial Flanks.- Another modification 
which may be made is to have the teeth of both 
gears with radial flanks. If, for instance, in Fig. 
161 a generating circle were to be used in the 










196 SELF-TAUGHT MECHANICAL DRAWING 

lower portion, of half the pitch diameter of the 
large gear, the effect would be to give to that gear 
radial flanks, and to make the points of the teeth 
of the small-gear broader in order to work properly 
with them. Then both gears would have radial 
flanks. Such gears have been considerably used. 
They are not as strong as gears of the standard 
shape, and the only advantage is that it is easier 
to make the pattern, the teeth being all worked out 
with a flat-faced plane; but as the teeth of in¬ 
volute gears, described in the next section, can be 
worked out in the same way, and as such gears are 
interchangeable, the advantage is obviously in 
favor of the involute system for such work. 

Involute Gears.—In involute gears the working 
surfaces of the teeth are involutes, formed not 
upon the pitch circles, but upon base circles lying 
within the pitch circles and tangent to a line, 
called the line of action, which passes obliquely 
through the point where the pitch circles cross the 
line connecting their centers. Let a and h, Fig. 
159, be pitch circles, and let the line cd be the line 
of action. Then e and /, being made tangent to 
the line cd, will be the base circles upon which 
the involutes are to be formed. If now this line 
of action be considered as part of a thread which 
unwinds from one base circle and winds up on the 
other, as the pitch circles are revolved back and 
forth upon each other, then if tracing points were 
attached to the thread at points 1, 2, S, Uy 5 and 6, 
these points would describe involutes outwardly 
from the base circles, which, being formed simul¬ 
taneously in pairs and each pair being formed by 


GENERAL PRINCIPLES OF GEARING 197 

a common point, would work together smoothly 
like those formed by the generating circles of the 
epicycloidal system. That the base circles are of 
such size as to just pass the thread as the pitch 
circles roll upon each other is proven by the fact 
that their radii, gd and gi, and he and hi, the radii 
gd and he being made at right angles to the line of 
action, are corresponding sides of similar triangles, 
the segments into which the line of action is di¬ 
vided by the line of centers being the other sides, 
and hence have the same ratio. It would only then 



Fig. 165.—Modified Form of Involute Rack Teeth. 


be necessary to reverse the direction of the thread 
to get curves for the other side of the teeth, and 
to give to the teeth their proper length inside and 
outside of the pitch line to obtain a pair of cor¬ 
rectly working involute gears. That part of the 
tooth of an involute gear which may lie within the 
base line is made radial. 

In the standard interchangeable involute gears 
the line of action is given an obliquity of 15 
degrees (cut gears, 14J degrees). This angle may 
be readily obtained by the combination of the 
triangles resting against the blade of the T-square 
shown in Fig. 166. The point of contact of the 











198 SELF-TAUGHT MECHANICAL DRAWING 

teeth is always upon the line of action and the 
push of one tooth against another is in its direc¬ 
tion, hence its name. 

The teeth of the 15-degree involute rack have 
straight sides, inclined to the pitch line at an angle 
of 75 degrees as shown in Fig. 164. This shape, 
however, is subject to a slight modification to avoid 
interference of the points of the teeth with the 
radial fianks of small gears. 

Interference in Involute Gears.—The points c 
and d, Fig. 159, where the line of action is tangent 
to the base circles, are called the limiting points. 
If the involutes which spring from either base cir¬ 
cle are so long as to reach 
beyond these limits on the 
other base circle, they will 
interfere with the radial 
fianks of the mating teeth. 
At k is shown an elongated 
involute interfering with 
the radial fiank of the 
mating tooth. This is, of 
course, a highly exagger¬ 
ated case. The interfer¬ 
ence will occur sooner as the line of action is made 
to cross the line of centers at a less oblique angle, 
as in standard gears, and still earlier as the pitch 
circle b is made larger. In gearing of standard pro¬ 
portions, a gear of 30 teeth is the smallest that will 
work correctly with a straight toothed rack. In 
the gears shown in Fig. 163, the teeth of the large 
gear pass beyond the limiting point of the small 
gear, and hence, if made of true involute shape. 



Fig. 166.—Obtaining a 15- 
or 75-degree Angle by 
30- and 45-degree Tri¬ 
angles. 








GENERAL PRINCIPLES OF GEARING 199 

their extremities will not work properly with the 
flanks of the small gear. 

There are three methods available to overcome 
this interference. First, to hollow out the flanks 
of the teeth of the small gear. Second, to round 
off the points of the teeth of the large gear. This 
is the method usually adopted, in interchangeable 
gears, the point being rounded off enough to clear 
the flanks of the smallest gear of the set. Fig. 165 
shows the teeth of the rack so corrected in larger 
scale. Third, to cut off that part of the tooth in 
the large gear which extends beyond the limiting 
point of the small gear. This is done in special 
cases. 

The Two Systems Compared.—The great point 
in favor of epicycloidal gearing would appear to be 
in its freedom from interference. It is necessary, 
however, in order to have epicycloidal gears run 
well, to have the pitch circles of the two gears of 
a pair just coincide, as shown in Fig. 161; but 
with involute gears the distance between centers 
may be varied somewhat without affecting their 
smoothness of operation, though where the points 
of the teeth are rounded off to avoid interference, 
as previously explained, the amount of variation 
which can be allowed is not great. As no value 
has been given to the angle at which the line of 
action crosses the line of centers in Fig. 159, it is 
evident that whether the base circles are brought 
nearer together or are carried further apart, circles 
which might then be drawn through the point 
where the line of action crosses the line of centers, 
would roll upon each other while the base circles 


200 SELF-TAUGHT MECHANICAL DRAWING 


passed the thread as before, and hence would be 
true pitch circles for the time being. The amount 
of backlash, that is, the space between the faces 
of the teeth, would vary, but the smoothness of 
operation would not be affected. This property of 
involute gears is very valuable in cases where the 
distance between centers is variable, as in rolling 
mill gearing. In such cases, however, interfer¬ 
ence must be avoided by the first of the three 
methods explained, that of hollowing out the flanks 
of the teeth of the mating gear. 

The epicycloidal system is the older of the two, 
and cast gears are still quite largely made to this 
system, there being so many patterns of that sys¬ 
tem on hand. But though the epicycloidal system 
once had the field to itself, the fact that the invol¬ 
ute system has so largely replaced it, having al¬ 
most wholly superseded it for cut gearing, shows 
the trend of modern practice. It is sometimes 
urged against the involute system that the thrust 
on the shaft bearings is greater than with the epi¬ 
cycloidal system, on account of the obliquity of its 
line of action. But though the line of action is at 
an angle to the direction of the motion of the teeth 
when they are on the line connecting their centers, 
it is a constant angle; while it is never less, it is 
never more. With the epicycloidal system, on the 
other hand, though the teeth of the driver give a 
square push to the teeth of the driven gear when 
they are in contact on the line of centers, yet 
the direction of this pushing action being on the 
'line of the generating circle, is variable, so that 
when the teeth are first coming into contact with 


GENERAL PRINCIPLES OF GEARING 201 

one another they have an obliquity of action fully 
as great, if not greater, than standard involute 
gears. For this reason such authorities as the 
Brown & Sharp Co., Grant and Unwin, do not con¬ 
sider this objection as being of great weight. 

Twenty-Degree Involute Gears.—It has been al¬ 
ready shown how the teeth of epicycloidal gears 
may be considerably strengthened where it is not 
necessary to have them interchangeable. In invol¬ 
ute gearing, when a stronger gear is desired than 
the standard 15-degree tooth provides for, recourse 
may be had to increasing the obliquity of the line 
of action. This makes the tooth considerably 
broader at the base, and correspondingly narrower 
at the point. The angle usually adopted in such 
cases is 20 degrees, and some makers report an 
increasing demand for such gears. 

Shrouded Gears. — When it is desired to strengthen 
the teeth of cast gears without increasing their 
size, or without using any other than a standard 
shape or tooth, the practice of shrouding them is 
sometimes resorted to. This consists in casting 
a flange on one or both sides of the gear. Full 
shrouding consists in having the flanges extend to 
the points of the teeth as shown in Fig. 167; half 
shrouding is where the flanges extend only to the 
pitch line as shown in Fig. 168. When the two 
gears of a pair are of nearly equal size so that 
their teeth would be of about the same strength 
it would be natural to use half shrouding on both 
gears as shown. 

When, however, there is much difference in the 
size of the gears, as shown in Fig. 167, it would be 


202 SELF-TAUGHT MECHANICAL DRAWING 


natural to use full shrouding on the small gear, as 
otherwise its teeth would be weaker than those 
of the large gear. Shrouding is estimated to 
strengthen the teeth from 25 to 50 per cent. 




Figs. 167 and 168. —Shrouded Gears. 


Bevel Gears.—In cylindrical or spur gears the 
pitch surfaces are cylinders of a diameter equal to 
the pitch circle; in bevel gears the pitch surfaces 
are cones, having their apices coinciding. 

In designing a pair of bevel gears as shown in 
Fig. 169, the center lines ab and cd are first drawn, 
and the pitch diameters then laid out from these 












































GENERAL PRINCIPLES OF GEARING 


203 


lines as indicated. From the point where the lines 
of the pitch diameters meet at e, a line is drawn to 
the point where the center lines intersect at h. 
This gives one side of the pitch cone of each gear 
and from this the other sides of the cones are 



readily drawn. All lines of the working surfaces 
of the gears meet at the point h. 

To lay out the teeth, the line fg is first drawn 
through the point e and at right angles to eh. This 
gives the outside face of the teeth, and the points 
/and g become the apices of cones upon the devel¬ 
opment of which the teeth are laid out. With cen¬ 
ters at / and g the pitch line developments ei and 
ej are drawn, and upon these lines the teeth are 
laid out the same as for ordinary gears. When 
the two gears of a pair are of the same size 
they are called miter gears. 

























204 SELF-TAUGHT MECHANICAL DRAWING 


Worm Gearing.—In worm gearing, as shown in 
Fig. 170, a screw having its threads shaped like 
the teeth of a rack engages with the teeth of a 
gear having a concave face and teeth of such shape 
as to fit the threads of the screw. If the screw is 
single threaded, one rotation of it will cause the 
gear to revolve the distance of one tooth; if double 
threaded, the gear will turn two teeth, and so on. 

In worm gearing, the worm wears much faster 
than the gear; it is, therefore, frequently made of 



Fig. 170.—Worm and Worm-Gear. 


steel while the worm-wheel is made of bronze, to 
give the combination increased durability. 

In involute worm gearing interference is com¬ 
monly avoided by the last of the three methods 
already mentioned. The points of the thread of the 
screw in Fig. 170 project but little beyond the 
pitch line, the root spaces of the gear being made 
correspondingly shallow. At the same time, the 
points of the teeth in the gear are made long 
enough to preserve their total length the same as 
usual, and the depth of the screw thread inside the 
pitch line is made sufficient for clearance. But un- 




















GENERAL PRINCIPLES OF GEARING 205 

less the worm-gear has less than 30 teeth, the 
standard shape of tooth will be satisfactory. 

Circular Pitch.—In designing gearing, the old 
method (the one which is given in the older trea¬ 
tises on the subject) is to use the circular pitch; 
‘ that is, the distance between the teeth, measured 
from center to center on the pitch circle. This 
method has many disadvantages. For instance, if 
it is required to make a pattern of a gear to mesh 
with one already on hand, the natural thing to do 
in measuring up the old gear is to first guess at 
where the pitch line is, and then measure straight 
across from one tooth to the next. This leads to 
two errors in the result; first, the probably incor¬ 
rect location of the pitch line, and, second, the dis¬ 
tance measured is the chordal pitch instead of the 
circular pitch. A noisy pair of gears would quite 
likely be the result. 

Again, as the ratio between the circumference 
and the diameter of a circle is not an even num¬ 
ber, but a troublesome fraction, the use of the cir¬ 
cular pitch method will give the pitch diameter of 
the gear in inconvenient fractions of an inch, un¬ 
less an equally inconvenient circular pitch is used. 
This method has so many disadvantages that it 
has been largely replaced by the more convenient 
“diametral pitch^^ method. For cut gears the dia¬ 
metral pitch method is used almost exclusively; 
but for cast gears there are so many patterns on 
hand, made by the circular pitch method, that that 
method is still used considerably on such work, 
especially on the larger sizes of gears. 

Where one is designing new work, however. 


206 SELF-TAUGHT MECHANICAL DRAWING 

where no old gear patterns made by the circular 
pitch method are used, the diametral pitch method 
will be by far the most convenient to use, which¬ 
ever style of tooth, whether involute or epicy- 
cloidal, may be adopted. 

PITCH DIAMETERS OF GEARS FROM 10 TO 100 
TEETH, OF 1-INCH CIRCULAR PITCH. 


No. 

of 

Teeth 

Diam. in 
Inches 

No. 

of 

' Teeth 

Diam. in 
Inches 

No. 

of 

Teeth 

Diam. in 
Inches 

No. 

of 

Teeth 

Diam. in 
Inches 

10 

3.183 

33 

10.504 

56 

17.825 i 

79 

25.146 

11 

3.501 

34 

10.823 

57 

18.144 I 

80 

25.465 

12 

3.820 

35 

11.141 

58 

18.462 i 

81 

25.783 

13 

4.138 

36 

11.459 

59 

18.781 < 

82 

26.101 

14 

4.456 

37 

11.777 

60 

19.099 i 

83 

26.419 

15 

4.775 

38 

12.096 

61 

19.417 ! 

84 

26.738 

16 

5.093 

i 39 

12.414 

62 

19.735 * 

85 

27.056 

17 

5.411 

40 

12.732 

63 

20.054 s 

86 

27.375 

18 

5.730 

41 

13.051 

64 

20.372 i 

87 

27.693 

19 

6.048 

42 

13.369 

65 

20.690 

88 

28.011 

20 

6.366 

43 

13.687 

66 

21.008 

89 

28.329 

21 

6.685 

I 44 

14.006 

67 

21.327 

90 

28.648 

22 

7.003 

45 

14.324 

68 

21.645 

91 

28.966 

23 

7.321 

i 46 

14.642 

69 

21.963 

92 

29.285 

24 

7.639 

! 47 

14.961 

70 

22.282 

93 

29.603 

25 

7.958 

48 

15.279 

71 

22.600 

94 

29.921 

26 

8.276 

49 

15.597 

72 

22.918 

95 

30.239 

27 

8.594 

50 

15.915 

73 

23.236 

96 

30.558 

28 

8.913 

51 

16.234 

74 

23.555 

97 

30.876 

29 

9.231 

52 

16.552 

75 

23.873 

98 

31.194 

30 

9.549 

1 53 

16.870 

76 

24.192 

99 

31.512 

31 

9.868 ! 

54 

17.189 

77 

24.510 

100 

31.831 

32 

10.186 i 

55 

17.507 

78 

24.828 

i 



When the pitch of a gear is given in inches or 
fractions of an inch, the circular pitch is always 
meant; as, for instance, where a gear is said to be 
of 1-inch pitch, or IJ-inch pitch. To get the pitch 
diameter in such a case, it is necessary to multiply 


































GENERAL .PRINCIPLES OF GEARING 207 

this pitch by the number of teeth in the gear, and 
then divide this product by 3.1416, the ratio be¬ 
tween the circumference and the diameter. For 
ascertaining the pitch diameter of gears when 
using the circular pitch, the accompanying table 
will save much time. If the gear is of any other 
than 1-inch circular pitch, multiply the diameter 
here given for the required number of teeth, by 
the circular pitch to be used. 

Proportions of Teeth.—The proportions of the 
teeth of gears where the circular pitch method is 
used, are given slightly different by various writ¬ 
ers. The length of the teeth is entirely arbitrary 
and therefore this discrepancy is quite natural. 
It is also unimportant, excepting as uniformity 
is desirable. The proportions as given by Grant 
are as follows: The addendum and dedendum are 
each made one-third of the circular pitch; the 
clearance, the distance of the root line below the 
dedendum line, is made one-eighth of the adden¬ 
dum; the backlash, the space which is allowed be¬ 
tween the sides of the teeth in cast gears, is made 
about the same as the clearance. This presents the 
proportions in fractions which are convenient to 
use, and at the same time makes the proportions 
practically the same as those of the diametral pitch 
method. Cut gears are made without backlash. 

Diametral Pitch.—In the diametral pitch method 
the gear is considered as having a given number 
of teeth for each inch of pitch diameter. Gears 
having three, four, or five teeth to each inch of 
their pitch diameters are said to be of three, four, 
or five pitch. With this method the addendum 


208 SELF-TAUGHT MECHANICAL DRAWING 


(the distance which the teeth project beyond the 
pitch line) is made equal to one divided by the 
pitch, so that the addendum on gears of three, four 
or five pitch would be, respectively, one-third, one- 
fourth or one-fifth of an inch. The advantages of 
this method are numerous. 

To get the diametral pitch of a gear it is only 
necessary to divide the number of teeth by the 
pitch diameter, or to divide the number of teeth 
plus two, by the outside diameter. A complete 
set of rules, as well as formulas a'nd examples for 
calculating spur gear dimensions, will be given in 
the next chapter. 

It is quite a common practice in figuring gears 
made by diametral pitch to give only the pitch and 
the number of teeth, as 4 pitch, 18 teeth, or 4 D. 
P., 18 T. The letters D. P. stand for diametral 
pitch, the letters P. D. standing for pitch diameter. 
The pitch diameter is then found by dividing the 
number of teeth by the diametral pitch. When 
this method is used, the circular pitch becomes 
of secondary importance, but may be found by di¬ 
viding 3.1416 by the diametral pitch. When the 
circular pitch is given and the diametral pitch is 
desired, divide 3.1416 by the circular pitch. The 
diameter of a gear, unless otherwise specified, is 
always understood to be the pitch diameter. With 
the diametral pitch method, the pitch diameter, 
unless in even inches, will be in fractions of an 
inch corresponding to the pitch, so that the frac¬ 
tional parts of the diameter of gears of three, four 
or five pitch, for instance, would be thirds, fourths 
or fifths of an inch. 


GENERAL PRINCIPLES OF GEARING 


209 


The Hunting Tooth.—It is a common practice in 
making gear patterns to have the teeth of the two 
gears of a pair of such numbers that they do not 
have a common divisor. For instance, instead of 
having 25 and 35 teeth in the gears of a pair, one 
may give to one of them one more or one less 
tooth, so as to insure all of the teeth of one gear 
coming into contact with all of the teeth of the 
other as they run together. 

This practice is condemned by some, however, 
on the ground that if any of the teeth are of bad 
shape it would be better to confine their injurious 
action within as narrow limits as ‘possible, rather 
than to have them ruin all of the teeth of the other 
gear; but the shape of badly formed teeth should 
be corrected as soon as the error is discovered. 

Approximate Shapes for Cycloidal Gear Teeth.— 
That part of the cycloidal curve which is used in 
the formation of gear tooth outlines is so short 
that it may be replaced with a circular arc which 
will very closely approximate it, and such arcs are 
generally used in the practical construction of gear 
patterns. In the following is given a table of such 
arcs with the location of the centers from which 
they are struck. The center from which that part 
of the tooth lying outside of the pitch line is 
drawn, the face of the tooth, will be inside of the 
pitch line, while the center from which that part 
of the tooth lying inside of the pitch line is drawn, 
the flank of the tooth, will be outside of the pitch 
line. These radii and center locations were ob¬ 
tained directly from a set of tooth outlines of 
3-inch circular pitch, formed by rolling a genera- 


210 SELF-TAUGHT MECHANICAL DRAWING 


ting circle, drawn upon tracing paper, upon a set 
of pitch circles, correct rotation being assured by 
the use of needle points pricked through the gen¬ 
erating circle into the pitch circle, the needle 
points serving as pivots upon which the genera¬ 
ting circle was swung through short successive 
stages, the forward movements of the tracing 
point in forming the cycloidal curves being also 
pricked through. Needle points were also used in 
the instruments which were used for tracing this 
curve when the radius and center location were 
determined. 

CYCLOIDAL TOOTH OUTLINES 

Radii and center locations for one-inch circular pitch. For 
any other pitch multiply the given figure by the required 
pitch. 


Number of 
Teeth. 

Face 

Radius. 

Inside of j 
Pitch Line. 

Flank 

Radius. 

Outside of 
Pitch Line. 

12 

0.625 ins. 

0.016 ins. 

Radial 


14 

0.666 

i i 

0.021 

i i 

4.00 ins. 

2.^5 ins. 

16 

0.697 

i i 

0.026 

i i 

2.80 

i i 

1.33 “ 

18 

0.724 

i i 

0.031 

i i 

2.37 

i i 

0.96 “ 

20 

0.750 

i i 

0.036 


2.14 

i i 

0.73 “ 

25 

0.802 

i i 

0.042 

i i 

1.91 

i i 

0.58 

30 

0.844 

i < 

0.052 

i i 

1.79 

i i 

0.48 “ 

40 

0.906 

i i 

0.062 

i i 

1.64 

i < 

0.375 “ 

60 

0.958 

i i 

0.083 

i i 

1.50 

4 4 

0.29 “ 

100 

1.010 

i i 

0.095 

i i 1 

1.33 

4 4 

0.21 “ 

200 

1.040 

i i 

0.120 

i t 

1.23 

4 4 

0.177 “ 

Rack 

1.080 

i i 

0.127 

i i 

1.08 

4 4 

0.127 


If the diametral pitch method is being used, the corresponding circular 
pitch may be found by dividing 3.1416 by the diametral pitch, as already 
mentioned. 

Involute Teeth.—The construction of a correct 
involute tooth outline is so simple a matter as to 
make the use of tables of approximate circular 



























GENERAL PRINCIPLES OF GEARING 211 

arcs unnecessary. An involute may be formed by 
the plotting method given in the geometrical prob¬ 
lems, but in most cases it may be more readily 
formed by the use of a sharply pointed pencil 
guided by a strong thread as shown in Fig. 171, 
where ab represents the pitch line of a gear, and 
cd represents the base circle, having a number of 
pins stuck into it at short distances apart. The 
thread being doubled, forms a loop to hold the pen¬ 
cil point. The thread being drawn tightly around 
the pins, the pencil is swung outward from the 



Fig. 171.—Laying out an Involute Gear Tooth. 


base circle, forming the required involute. When 
gears of over thirty teeth are to mesh into others 
of less than that number, it will be necessary to 
slightly round over the points of the teeth to avoid 
interference with the radial flanks of the mating 
gear. For this purpose use a radius of 2.10 inches 
divided by the diametral pitch, with a center on 
the pitch line as shown in Fig. 172. This radius, 
2.10 inches divided by the diametral pitch, is the 
same as that given by Grant for rounding off the 
points of the teeth of racks; but actual trial on 
teeth of large size shows it to be correct for gear 
wheels also, giving a curve which coincides very 
closely with the epicycloidal shape which the point 




212 SELF-TAUGHT MECHANICAL DRAWING 

should have to work correctly with the radial flank 
of the mating gear. 

That part of an involute tooth lying within the 
base circle is made radial, as previously stated, and 
a good fillet should be drawn in at the root. For 
this purpose use a radius of one-twelfth of the 
circular pitch. A templet which is fitted to this 



Fig. 172.—Modified Tooth Form to Avoid Interference. 

outline is used to finish the drawing, and to mark 
out the teeth on the pattern. 

On large work the size of the base circle may be 
obtained by calculation more readily than by the 
use of the triangle, as shown in Fig. 166. When 
the line of action has an obliquity of 15 degrees, 
the diameter of the base circle will be equal to 
0.966 of the pitch diameter. For 20-degree invo¬ 
lute gears the diameter of the base circle will be 
0.94 of the pitch diameter. 

With the 20-degree involute system the teeth of 
the rack have an inclination of 70 degrees to the 
pitch line. With this system there will be no ne¬ 
cessity for rounding off the points of the teeth of 
the rack or of a large gear unless it meshes with 








GENERAL PRINCIPLES OF GEARING 213 

a gear of less than 18 teeth. When, to avoid inter¬ 
ference, it does become necessary to round off the 
points of the teeth of the rack or of large gears, 
the same radius, 2.10 inches divided by the diam¬ 
etral pitch, is to be used, as in the 15-degree 
system, the center being on the pitch line as 
before. 

Proportions of Gears.—A somewhat common rule 
is to make the rim and the arms of about the same 
thickness as the teeth at the root, though some 
make the thickness of the rim equal to the height 
of the tooth; and to make the diameter and length 
of the hub about equal to about twice the diameter 
of the shaft. On spoked gears, the rim is also stiff¬ 
ened by ribbing it between the arms. On a light 
gear mounted on a relatively large shaft it would 
be natural to lighten the hub somewhat. The 
width of Ahe face of cast gears is usually made 
from two to three times the circular pitch. The 
face of bevel gears should not exceed one-fifth of 
the diameter of the large gear, and the face of 
worm gears should not exceed one-half of the 
diameter of the worm. 

Strength of Gear Teeth.—When a gear is to be 
designed for a given work, the first question is 
how large to make the teeth to give the required 
strength. On their size will also depend the gen¬ 
eral proportions of the gear. 

It is comparatively easy to determine the work 
which the teeth are doing, that is, the strain or 
load which they are bearing, when the power 
which the gear transmits is known. A horse-power 
being the power required to lift 33,000 pounds one 


214 SELF-TAUGHT MECHANICAL DRAWING 

foot in one minute, the load on the teeth will be 
33,000 multiplied by the horse-power which is 
being transmitted, and divided by the velocity of 
the pitch line of the gear in feet per minute; or, 
what is the same thing, 126,050 multiplied by the 
horse-power, and divided by the product of the 
pitch diameter in inches multiplied by the number 
of revolutions per minute. This latter figure, 
126,050, takes into account the fact that in the first 
case the velocity is expressed in feet, while in this 
case the diameter is in inches, and also the fact 
that the velocity is a factor of the circumference 
instead of the diameter. 

While the load on the teeth may be readily 
determined, the question of how large they should 
be made to bear it is one where authorities have 
differed very much on account of the number of 
factors involved. First of all is the question of 
the material, usually cast iron, which is a variable . 
quantity, both on account of the nature of the 
material itself, different grades varying greatly as 
to strength, and the liability of defects in the cast¬ 
ing. Then there is the question of whether the 
load should be considered as divided between two 
or more teeth or carried by one tooth, or the cor¬ 
ner of a tooth. 

Then there is the nature of the work: whether 
the load will be uniform or whether the teeth will 
be subject to severe strain or shock. There are 
questions of the shape of the tooth, and the velocity 
at which the gear is running, the teeth having 
greater strength at slow speeds than at high speeds 
due to the shocks accompanying high velocities. 


GENERAL PRINCIPLES OF GEARING 215 

To show the different results given by different 
writers we may take the case of a gear 24 inches 
diameter, 2 inches circular pitch, 4 inches face, 
running at 100 revolutions per minute. A rule 
given by Box in his treatise on mill gearing, and 
quoted by Grant and Kent, would make the gear 
safe for 9.4 horse-power. The rule in Nystrom’s 
Mechanics gives 12.2 horse-power. Rules by other 
writers, quoted by Kent, give results as follows: 
Halsey, 22.6; Jones & Laughlin, 35; Harkness, 38; 
Lewis, 65.2. The rule by Prof. Harkness is the 
result of investigations conducted by him in 1886. 
He examined a great many rules, largely, how¬ 
ever, for common cast gears. Mr. Lewis’s method, 
the result of his investigations of modern machine 
molded and cut gears, though giving much higher 
results than the others, is said to have proved sat¬ 
isfactory in an extensive practice, and so miay be 
considered reliable for gears which are so well 
made that the pressure bears along the face of the 
teeth instead of upon the corners. 

It is customary in calculating gears to proceed 
on the assumption that the load is borne by one 
tooth, and in ordinary work, the size of the tooth 
may be determined by the load it may safely bear 
per inch of face and per inch of circular pitch. 

In 1879, J. H. Cooper selected an old English rule 
giving the breaking load of the tooth as 2000 X 
pitch X face, which, allowing a factor of safety of 
10, would give us a safe load of 200 X pitch X face. 
Kent says of this rule that for rough ordinary work 
it “is probably as good as any, except that the fig¬ 
ure 200 may be too high for weak forms of tooth. 


216 SELF-TAUGHT MECHANICAL DRAWING 


and for high speeds. ” Lewis also considers this 
rule as a passably correct expression of good gen¬ 
eral averages. 

The value given by Nystrom and those given by 
Box for teeth of small pitch, are so much smaller 
than those of other authorities that Kent says they 
may be rejected as giving unnecessary strength. 
Accepting the factor 200 as a good average would 
leave one room for the exercise of individual judg¬ 
ment for the particular case in hand. If the speed 
were slow and the teeth were of strong shape, as 
where both the gears of a pair, or all of the gears 
of a train, have a reasonably large number of 
teeth, a higher figure, perhaps 225 or more, might 
be taken; while if the speed were higher and one 
of the gears had but few teeth, giving them a 
weak form, or if they were to be subject to much 
vibration or shock, a lower figure, perhaps as low 
as 125, might be taken. 

To ascertain the horse-power safely transmitted 
by an existing gear, we would then multiply to¬ 
gether its diameter, pitch (circular) and face, 
taken in inches, and the number of revolutions 
per minute, and multiply their product by 200, or 
whatever figure is selected, and divide the total 
product by 126,050. This may, perhaps, be ex¬ 
pressed clearer, as follows: 

TT diam. X rev. X circ. pitch X face X 200 

Horse-power=-- 

The figure 200 would give to the 24-inch gear 
previously considered 30.5 horse-power. The fig¬ 
ure 125 would give 19.0 horse-power. 




GENERAL PRINCIPLES OF GEARING 


217 


To ascertain the size of the teeth to transmit a 
given horse-power we may transpose the above rule 
and say that the product of the pitch multiplied 
by the face would be equal to 126,050 multiplied 
by the horse-power, and divided by the product of 
the diameter in inches, the number of revolutions 
per minute, and 200, or the figure selected; that is: 


Circ. pitch X face 


126,05 0 X h orse-power 
diam. X rev. X 200. 


Assuming some pitch and dividing this result 
by it would give the breadth of face. A few 
trials will give the desired ratio between pitch 
and breadth of face. If one has a table of square 
roots at hand, the work may be simplified by 
assuming some desired ratio, when the pitch will 
be the square root of the quotient of this figure, 
pitch multiplied by the face, divided by the ratio. 
If, for instance, the pitch multiplied by the face 
were found to be 12, and we desired them to be 
in the ratio of 24 to 1, the pitch would be equal to 
the square root of the quotient of 12 divided by 
24, or 2.191, which would be about the same as 14 
diametral pitch. 

Example .—Required the size of the teeth of a 
gear 18 inches in diameter, to run 120 revolutions 
per minute, which shall transmit five horse-power, 
allowing 200 pounds load per inch of face, and 
inch of pitch. Then: 


Pitch X face = 


126,050 X 5 
18 X 120 X 200 


680,250 

432,000 


1.46 


nearly. A circular pitch of 0.785 inch, correspond- 




218 SELF-TAUGHT MECHANICAL DRAWING 


ing* to 4 diametral pitch, would give a breadth of 
face of about li inches. For bevel gears take the 
diameter and pitch at the middle of the face. 

Mr. Lewis’s method differs from the preceding 
in that instead of using a single constant, as 200 
pounds per inch of pitch and inch of face, two 
constants are used, one, Y, a factor of strength 
depending on the number of teeth in the gear, and 
another, S, a safe working stress for different 
speeds of the pitch line, in feet per minute. The 
values of these constants are given in the accom¬ 
panying tables. 

The rule to get the horse-power of a given gear 
is: 

xj p _ circ. p itch X face X veloc ity X S X Y 
^ “ 33,000 


the velocity being that at the pitch line in feet per 
minute, and the values of S and Y being taken 
from the tables. The velocity is, of course, the 
diameter in feet X 3.1416 X number of revolu- 
tions. If the diameter were taken in inches then 
the total product would be divided by 12. The 
product of the pitch multiplied by the face, to 
determine the size of teeth to transmit a given 
power, would then be 


Circ. pitch X face 


_33,000J< H._P._ 
velocity X S X Y. 


The calculation should be made for the gear of 
the pair or train having the fewest teeth, as it 
would be the weakest, unless it were made of some 
stronger material as steel, or unless it were 







GENERAL PRINCIPLES OF GEARING 


219 


WORKING STRESS, S, FOR DIFFERENT SPEEDS 
AT PITCH LINE IN FEET PER MINUTE, 

FOR CAST IRON. 


Speed. 

s. 

Speetl. 

s. 

100 or less 

8000 

900 

3000 

200 

6000 

1200 

2400 

300 

4800 

1800 

2000 

600 

4000 

2400 

1700 


shrouded. If made of steel S might be taken 2i 
times the tabulated values. 

As a gear with cut teeth has from two to three 
times the strength of one with cast teeth, because 
of the more perfect contact, Mr. Lewises method 
might be adapted to common cast gears by taking 
the value of S at from one-half to one-third of the 
tabulated value. By so doing one could bring into 
the calculation the question of shape of teeth and 

FACTOR FOR STRENGTH, Y, TO BE USED IN 
LEWIS’S FORMULAS. 


No. of 
teeth. 

20 degree 
involute 

15 degree 
involute 
and 

cycloidal. 

No. of 
teeth. 

20 degree 
involute 

15 degree 
involute 
and 

cycloidal. 

1 

No. of 
teeth 

20 degree 
involute. 

15 degree 
involute 
and 

cycloidal 

12 

0.078 

0.067 

20 

0.102 

0.090 

43 

0.126 

0.110 

13 

0.083 

0.070 

21 

0.104 

0.092 

50 

0T30 

0.112 

14 

0.088 

0.072 

23 

0.106 

0.094 

60 

0.134 

0.114 

15 

0.092 

0.075 

25 

0.108 

0.097 

75 

0.138 

0.116 

16 

0.094 

0.077 

1 27 

0.111 

0.100 

100 

0.142 

0.118 

17 

0.096 

0.080 

30 

0.114 

0.102 

150 

0.146 

0.120 

18 

0.098 

0.083 

34 

0.118 

0.104 

300 

0.150 

0.122 

19 

0.100 

0.087 

38 

0.122 

0.107 

Rack 

0.154 

0.124 














































220 SELF-TAUGHT MECHANICAL DRAWING * 


speed, which would be especially desirable if the 
speed were high or the teeth of weak form. Tak¬ 
ing S at one-half the tabulated value would give 
to the 24-inch gear previously considered about 
the same power as allowing 200 pounds per inch 
of pitch and face, which Mr. Lewis considers a 
fair value. With cast gears where interchange- 
ability is not a necessary feature, the teeth of a 
small gear could of course be considerably strength¬ 
ened in the manner previously indicated for epicy- 
cloidal gears; or the 20-degree system might be 
used if the teeth have the involute form. 

Thurston's Rule for Shafts.—The size of shaft 
which the gear will require may be found by the 
rule given by Thurston. Multiply the horse-power 
to be transmitted by 125 for iron, or by 75 for cold 
rolled iron, and divide the product by the number 
of revolutions per minute. The cube root of the 
quotient will be the size of the shaft. 

The size of gear to give a required speed may 
be readily determined from the fact that the prod¬ 
uct of the speed of the driving shaft multiplied by 
the size of the driving gear or gears, should be 
equal to the product of the speed of the driven 
shaft, multiplied by the size of the driven gear or 
gears. This, perhaps, may be made clearer by 
placing the driving members on one side of a line, 
and the driven members on the other side, as in 
the following example. 

A shaft making 75 turns per minute has on it a 
gear of 200 teeth. Required the size of gear to 
mesh with it which shall drive its shaft 120 


GENERAL PRINCIPLES OF GEARING 221 

revolutions per minute. Letting x represent the 
size of the required gear we have 

Rev. driving shaft = 75 ix = size driven gear. 

Size driving gear = 200 , 120 == rev. driven shaft. 

Then as the product of the numbers on one side 
of the line equals the product of those on the other 
side, 75 X 200 120 will give the value of x, the 

number of teeth in the driven gear. This method 
applies to a train of gears as well as a pair. 


I 


CHAPTER XIII 


CALCULATING THE DIMENSIONS OF GEARS 


In the previous chapter, the general principles 
of gearing have been explained. The three kinds 
of gearing most commonly in use, spur gearing, 
bevel gearing and worm gearing, have been 
touched upon, and the fundamental rules for the 
dimensions of gear teeth have been given. In 
this chapter it is proposed to give in detail the 
rules and formulas for these three classes of gears, 
so as to enable the student to calculate for himself 
any general problem in gearing with which he 
may meet. 

Spur Gearing.—In the following, machine cut 
gearing is, in particular, referred to; but the gen¬ 
eral formulas are, of course, of equal value for use 
when calculating cast gears. The expressions pitch 
diameter, diametral pitch and circular pitch have 
already been explained, and rules have been given 
for transferring circular pitch into diametral 
pitch, and vice versa. These rules, expressed as 
formulas, would be: 


P = 


3.1416 

p/ , 


and P' 


3H416 

P 


in which P = diametral pitch, and 
P'= circular pitch. 

Assume as an example that the diametral pitch 

^ 




CALCULATING THE DIMENSIONS OF GEARS 223 

of a gear is 4. What would be the circular pitch 
of this gear? 

Using the formula given, we have: 

p, ^ ^ Q 

4 

When the diametral pitch and the pitch diameter 
are known, the number of teeth may be found by 
multiplying the pitch diameter by the diametral 
pitch, as already mentioned in the previous chap¬ 
ter. This rule, expressed as a formula, would be: 

N = PXD 

in which = number of teeth, 

D = pitch diameter, and 
P = diametral pitch. 

Assume that the diametral pitch of a gear is 4 
and the pitch diameter 6i inches. What would be 
the number of teeth in this gear? 

By inserting the given values in the formula 
above, we would have: 

W = 4 X 6i = 25 teeth. 

If the number of teeth and pitch diameter of the 
gear are known, and the diametral pitch is to be 
found, a rule and formula for this may be arrived 
at by merely transposing the rule and formula just 
given. The diametral pitch equals the number of 
teeth divided by the pitch diameter, or, expressed 
as a formula : 



in which P, N and D signify the same quantities 
as in the previous formula. 



224 SELF-TAUGHT MECHANICAL DRAWING 


Assume, for an example, that the number of 
teeth in a gear equals 35 and that the pitch diam¬ 
eter is 3i inches. What is the diametral pitch? 

If we insert the known values in the given for¬ 
mula, we have: 

P = oi = diametral pitch. 

02 


Finally, if the diametral pitch and the number 
of teeth are known, the pitch diameter is found 
by dividing the number of teeth by the diametral 
pitch, which rule expressed as a formula, would be: 


As an example, assume that the number of teeth 
in a gear is 58 and the diametral pitch 6. What is 
the pitch diameter of this gear? 

By inserting the known values in the formula, 
we find: 

D = = 9.667 inches, 

b 


If it now be required to find the outside diam¬ 
eter of the gear, that is, the diameter of the gear 
blank, we make use of the following rule: The 
outside diameter equals the number of teeth plus 
2, divided by the diametral pitch. Expressed as 
a formula, this rule is: 


in which D' = outside diameter of gear, and N 
and P have the same significance as before. 

As an example, assume that the number of teeth 


CALCULATING THE DIMENSIONS OF GEARS 225 


is 58 and the diametral pitch 6. By inserting these 
values in the formula, we find the outside diameter: 


T)'= ^ 

6 



10 inches. 


When the pitch diameter and the diametral pitch 
are known, the outside diameter is found as 
follows: Add the quotient of 2 divided by the 
diametral pitch to the pitch diameter; the sum is 
the outside diameter. This rule, expressed as a 
formula, is: 

• D + ~ 


in which the letters have the same significance as 
before. 

Assume that the pitch diameter of a gear is 9.667 
inches, and the diametral pitch 6. Find the out¬ 
side diameter. 

By inserting the given values in the formula, we 
have: 

D'= 9.667 + = 9.667 + 0.333 = 10 inches. 

D 

By a transposition of the rule and formula just 
given, we find that the pitch diameter equals the 
outside diameter minus the quotient of 2 divided 
by the diametral pitch. This rule, written as a 
formula, is* 

D.iy-j; 

Assume that the diametral pitch of a gear is 8, 
and the outside diameter 12 inches. What is the 
pitch diameter? 

i) = 12 12 - i = Ilf inches. 

o 





226 SELF-TAUGHT MECHANICAL DRAWING 


When the number of teeth and outside diameter 
are known, the diametral pitch may be found by 
adding 2 to the number of teeth and dividing the 
sum by the outside diameter; or, expressed as a 
formula: 

p ^ iV+J 

^ D'. 


If the number of teeth in a gear is 96 and the 
outside diameter is 14 inches, what is the diame¬ 
tral pitch? 

If the known values are inserted in the given 
formula, we have: 


P 


96 + 2 98 


14 


14 


= 7 diametral pitch. 


When the outside diameter and the number of 
teeth are known, the pitch diameter may be found 
by multiplying the outside diameter by the number 
of teeth, and dividing the product by the sum of 
2 added to the number of teeth; or, as a formula: 

n = N 
AT + 2. 


Find the pitch diameter for the gear having 96 
teeth and an outside diameter of 14 inches. 


14^96 _ 1344 
96 + 2 98 


13.714 inches. 


When it is required to find the center distance C 
between two gears in mesh with each other, we 
must first know the pitch diameters of, or the 
number of teeth in, the two gears. The center 







CALCULATING THE DIMENSIONS OF GEARS 227 


distance equals one-half of the sum of the pitch 
diameters of the two gears: 

^ _ D -\~ d 


in which D and d denote the pitch diameters in 
the large and small meshing gears, respectively. 

The pitch diameters of two gears equal 9.5 and 7 
inches, respectively. Find the center distance 
between them when in mesh. 



9.5_4- 7 
2 


16.5 

2 


8.25 inches. 


The center distance is also equal to the sum of 
the numbers of teeth in the two gears divided by 
two times the diametral pitch; or, as a formula: 


in which N and n denote the numbers of teeth in 
the meshing gears. 

As an example, assume that the number of teeth 
in each two gears equals 95 and 75. The diametral 
pitch is 10. What is the center distance? 


95 + 75 
2 X 10 


170 

20 


= 8.5 inches. 


We will now find the dimensions of the tooth 
parts. The addendum (see Fig. 160) equals 1 
divided by the diametral pitch. Expressed as a 


formula: 



in which A = addendum. 



228 SELF-TAUGHT MECHANICAL DRAWING 

What is the addendum or height above the pitch 
line of a 5 diametral pitch gear tooth? 

A = 0.2 inch. 

5 

The dedendum (see Fig. 160) equals the ad¬ 
dendum. 

The clearance, c, equals 0.157 divided by the 
diametral pitch, or: 

a 157 
^ P. 

What is the clearance at the bottom of the gear 
tooth (see Fig. 160) of a 4 diametral pitch gear? 

c = = 0.039 inch. 

The full depth of the tooth equals the sum of the 
addendum, dedendum, and clearance, or 

-L 4- 4- 

^ p ' p ^ p p 

in which d' = full depth of gear tooth. 

What is the full depth of a 4 diametral pitch 
tooth ? 

d'= - f - = 0.539 inch. 

4 

The thickness of a cut gear tooth at the pitch 
line equals 1.5708 divided by the diametral pitch; 
or, as a formula: 

^ _ 1.5708 
P 


in which T = thickness of tooth at pitch line. 



CALCULATING THE DIMENSIONS OF GEARS 229 


What is the thickness at the pitch line of a 4 
diametral pitch gear tooth? 

T = = 0.3927 inch. 

4 


As a general example, let it be required to de¬ 
termine the various dimensions for a pair of gears, 
the one having 36 and the other 27 teeth. The 
gears are of 8 diametral pitch. 

By using the formulas given, we have: 

For the larger gear: 


Pitch diameter = 


Outside diameter = 


N ^ 
P 

AT + 2 
P 


= 4.5 inches. 


36 
8 

36+2 


8 


= 4.75 inches. 


For the smaller gear: 

n_ _ 21 
P " 8 

n + 2^27 + 2 
P 8 

For both gears: 

1 1 


Pitch diameter = 
Outside diameter= 


r = 3.375 inches. 

= 3.625 inches. 


Addendum = 


P 


8 


0.125 inch. 


Dedendum "" “p ^ "" 0.125 inch. 

Clearance = ^ 0.0196 inch. 


2.157 2.157 


Full depth of tooth = 'p g 

N+n 36 + 27 63 


0.2696 inch. 


Center distance = 


2P - 2 X 8 = 16 = 







230 SELF-TAUGHT MECHANICAL DRAWING 


This concludes the required calculations neces¬ 
sary for a pair of spur gears. 

Bevel Gears.—Bevel gears are used for trans¬ 
mitting motion between shafts whose shafts are 
not parallel, but whose center lines form an angle 
with each other. In most cases this angle is a 



Fig. 173.—Diagram for Calculation of Bevel Gearing. 

right, or 90-degree, angle. The formulas for the 
dimensions of bevel gears are not as simple as 
those for spur gears, and an understanding of the 
trigonometrical functions, explained in Chapter 
VII, is necessary, as well as the use of trigonomet¬ 
rical tables. As bevel gears with a 90-degree angle 
between their center lines are the most common, 






















































CALCULATING THE DIMENSIONS OF GEARS 231 


formulas will be given for this case only, in the 
following. 

In Fig. 173 a pair of bevel gears are shown, the 
dimensions of which are to be determined. The 
letters in the formulas below denote the following 
quantities: 

P = diametral pitch, 

Di = pitch diameter of large gear, 

. D 2 = pitch diameter of small gear, 

Oi = outside diameter of large gear, 

O 2 = outside diameter of small gear, 

Ni = number of teeth in large gear, 

N 2 = number of teeth in small gear, 

N/ = number of teeth for which to select cut¬ 
ter for large gear, 

A^ 2 '= number of teeth for which to select cut¬ 
ter for small gear, 

tti, 61 , Cl, a 2 , 62 , C 2 , d and e = angles as shown in 
Fig. 173. 

A = addendum, 

A + C = dedendum = addendum plus clearance. 
If the pitch diameter and diametral pitch are 
known, the number of teeth equals the pitch 
diameter multiplied by the diametral pitch, or: 

N, = D,XP 


N2 = D 2 X P 

If the number of teeth and the diametral pitch 
are known, the pitch diameter equals the number 
of teeth divided by the diametral pitch, or: 


D2 = 


p 

N2 

p 


232 SELF-TAUGHT MECH'ANICAL DRAWING 


Angles and aj can be determined if either the 
numbers of teeth or the pitch diameters of both 
gears are known. The tangent for these angles, 
the pitch cone angles, equals the number of teeth 
in one gear divided by the number of teeth in the 
other, or the pitch diameter in one gear divided by 
the pitch diameter in the other, according to the 
following formulas: 


tan ai = 


N, 

N, 


D, 

~D, 


0 


tan a 2 = 


Nj 

N, 



Angle Ua also equals 90° — a^. 

The outside diameter equals the pitch diameter 
plus the quotient of 2 times the cosine of or as, 
respectively, divided by the diametral pitch, or: 


Oi — Di + 


2 cos_Ui 

~ P 





2 cqs_a3 
P 


Angles d and e are determined by the formulas: 


, j 2 sin a, 2 sin a-, 
tan d = —TT— = XT “ 

No 


tan e = 


2.314 sin a, 2.314 sin a-. 


N, N 2 

Angles bi, Ci, 62 and C 2 are determined by the 
formulas: 

hi = ai + d 
Cl = ai — e 


62 — 0^2 P d 

Co - do C 





CALCULATING THE DIMENSIONS OF GEARS 233 


The number of teeth for which the cutter for 
cutting the teeth should be selected is found as 
follows: 

COS a I 



cos a 2 


Finally the addendum, dedendum and clearance 
are found as in spur gears. 

As a practical example, assume now that two 
bevel gears are required, 8 diametral pitch, with 
24 and 36 teeth, respectively. Find the various 
dimensions. 


Di = = Q =4.5 inches. 

X o 


N, 24 


D^- n g 


tan Ui = 
tan a 2 = 


P 

_ 36 
N, 24 

Ni ^ 24 
N, 36 


= 3 inches. 

= 1.5; = 56° 20'. 

= 0.667; a 2 = 33° 40'. 


Oi= Di-\- 

O2— 7)2 + 


2 cos Ui 

P 

2cos_a2 
• P 


4 ^ , 2 X 0.554 4 • 1 

= 4.5+ . - Q -- = 4.638 inches. 

o 

„ ,2 X 0.832 „ . , 

= 3 +- 5 -= 3.208 inches. 

O 


, 2 sin a, 2 X 0.832 ^ 7 

tan c? = —=--= 0.046; c? = 2° 40'. 

ob 

. 2.314 sin ai 2.314X 0.832 ^ 00 n/ 

tan c = -^- =- ^ -= 0.053; e =3 0 . 










234 SELF-TAUGHT MECHANICAL DRAWING 


6 , = ai + d = 56° 20' + 2° 40' = 59° O'. 

Cl = - e = 56° 20' - 3°0 ' = 53° 20'. 

^2 = as + d = 33° 40' + 2° 40' = 36° 20'. 

C3 = a2- c = 33°40'- 3°0' = 30°40'. 

N/= —= 65 approximately, 
cos ai 0.554 

N 2 = —A = 29 approximately, 
cos a 2 0.832 

A = —^ = 0.125 inch. 

Jr o 

C = - J— = 0.0196 inch. 

O 

Whole depth of tooth = -^ + p + = 

0.2696 inch. 

Worm Gearing.—Worms and worm gears are 
used for transmitting power in cases where great 



reduction in velocity and smoothness of action are 
desired. They are also used when a self-locking 




























CALCULATING THE DIMENSIONS OF GEARS 235 

power transmission is desirable, that is, when 
it is required that the mechanism itself, due to 
the friction between the worm and worm-wheel, 
should support the load without slipping if the 
driving power be rendered inoperative. 

In Figs. 174 is shown a worm and in Fig. 175 a 
worm-wheel; the dimensions to be found are, in 
most cases, given in these illustrations. The fol¬ 
lowing notation has been used in the formulas 
given below for worm and worm-wheels: 

P = circular pitch of worm-wheel = pitch of the 
worm thread, 

N = number of teeth in worm-wheel, 

Di = pitch diameter of worm-wheel, 

Dt = throat diameter of worm-wheel, 

Oi = outside diameter of worm-wheel (to sharp 
corners), 

R = radius of worm-wheel throat, 

C = center distance between worm and worm- 
wheel axes, 

D 2 = pitch diameter of worm, 

O 2 = outside diameter of worm. 

Dr = root diameter of worm, 

A = addendum, or height of worm tooth above 
pitch line, 

d = depth of worm tooth, 
a = face angle of worm-wheel. 

If the pitch of the worm and the number of 
teeth in the worm-wheel are known, the pitch 
diameter of the worm-wheel may be found by 
multiplying the pitch of the worm by the number 


23G SELF-TAUGHT MECHANICAL DRAWING 


of teeth, and dividing the result by 3.1416, or, as 
a formula: 

3.1416 

The outside diameter of the worm, O 2 , is usually 
assumed. To find the pitch diameter of the worm, 

the addendum must first 
be found. The addendum 
equals the pitch of the 
worm thread multiplied 
by 0.3183, or: 

A = PX 0.3183. 

Now the pitch diameter 
of the worm equals the 
outside diameter minus 2 
times the addendum, or: 

Da = O 2 - 2A. 

The root diameter of 
the worm can be found 
first after the full depth 
of the worm-wheel thread 
has been found. The full 
depth of the worm-wheel 
thread equals the pitch 
multiplied by 0.6866, or: 

d = P X 0.6866. 

Now the root diameter 
of the worm thread equals 
the outside diameter of 
the worm minus 2 times the depth of the thread, or: 

Dr = 0:> — 2d. 



























































CALCULATING THE DIMENSIONS OF GEARS 237 


The throat diameter of the worm-wheel is found 
by adding 2 times the addendum of the worm 
thread to the pitch diameter of the worm-wheel, or: 

Dt — + 2.A. 

The radius of the worm-wheel throat is found 
by subtracting 2 times the addendum from the 
outside diameter of the worm divided by 2, or: 



2A. 


The outside diameter of the worm-wheel (to 
sharp corners) is found by the formula below: 

0, = Dt + 2 (E - B cos -j) 

The angle a is usually 75 degrees. 

Finally, the center distance between the center 
of the worm and the center of the worm-wheel 
equals the sum of the pitch diameter of the worm 
plus the pitch diameter of the worm gear, and this 
sum divided by 2, or: 


Find, for an example, the required dimensions 
for a worm and worm-wheel, in which the worm- 
wheel has 36 teeth, the pitch of the worm thread 
is 4 inch, and the outside diameter of the worm is 
3 inches. We have given P = i; iV = 36; O 2 = 3. 



PXN _ 4 X 36 
3.1416 3.1416 


5.730 inches. 


A = PX 0.3183 = 4 X 0.3183 = 0.15915 inch. 
D2-0,~2A^3- 0,3183 = 2.6817 inches, 






238 SELF-TAUGHT MECHANICAL DRAWING 


d = 
Dr = 
Dt ~ 

R = 

(1.1817 
C = 


P X 0.6866 = i X 0.6866 = 0.3433 inch. 

O 2 - 2c^ = 3 - 0.6866 = 2.3134 inches. 

D, + 2A = 5.730 + 0.3183 = 6.0483 inches. 

^ - 2 A = Y - 0.3183 - 1.1817 inch. 

Dt + 2(R - R cos -|-) = 6.0483 + 2 X 
- 1.1817 X cos 37° 300 = 6.5375 inches. 


Di + D 2 

2 


5.730 + 2.6817 

2 


4.2058 inches. 




CHAPTER XIV 


CONE PULLEYS 


When it is desired to have a variable speed ratio 
between two shafts which are belted together, the 
method of having reversed conical cylinders or 
drums mounted on the shafts, as shown in Fig. 176 
and 177, is sometimes used. These permit any 



Fig. 176.—Simplest 
Form of ‘ ‘ Cone- 
Pulley.” 


Fig. 177.—An Im¬ 
proved Form of 
“Cone-Pulley.” 


Fig. 178.—The Mod¬ 
ern Type of Stepped 
Cone Pulley. 


desired change of speed, but they have disadvan¬ 
tages which on most work offset this advantage. 
It would be necessary, in the first place, to use a 
narrow belt to avoid undue stretching at the edges. 
Then, as the tendency of a belt is to mount to the 
largest part of a pulley, this tendency, acting in 

239 




























































240 SELF-TAUGHT MECHANICAL DRAWING 

the same way on the cones, would produce undue 
tension on the belt. If a crossed belt is used on 
such cones their faces would be made straight, as 
the belt would be equally tight in any position. 
This may be seen by an inspection of Fig. 179, 
where circles A and B represent sections of such 
cones on one line, and circles C and D represent 
sections on another line. If the cones have the 



Fig. 179.—Diagram Showing relative Influence of Open and 
Crossed Belt on Pulley Sizes. 


same taper it is evident that the circle D will be 
as much larger than 5 as C is smaller than A, the 
gain in one diameter being offset by the loss in the 
other. Then, as the circumferences of circles vary 
directly as their diameters (the circumference of 
a circle having twice the diameter of another, for 
instance, will be twice as long as the circumfer¬ 
ence of the other), whatever is gained on one cir¬ 
cumference will be lost on the other. For a crossed 
belt then, it is only necessary that the cones have 
the same taper. 

When, however, an open belt is used, it becomes 
necessary to have the cones slightly bulging in the 














CONE PULLEYS 


241 


middle as shown in Fig. 177. By again inspecting 
Fig. 179 it will be seen that it is only when the 
belt is crossed that one cone gains as fast in size 
as the other loses, because it is only when the belt 
is crossed that the arc of contact of the belt on the 
pulleys is the same on all steps of the cone. 

In practice these cones are usually replaced by 
stepped or cone pulleys as shown in Fig. 178, so as 
to avoid the troubles with the belt previously 
mentioned. 

Applying the principles mentioned to cone pul¬ 
leys, we see that when a crossed belt is used, all 
that is necessary is that the sum of the diameters 
of any pair of steps shall be equal to the sum of 
the diameters of any other pair of steps. For 
instance, the sum of the diameters of steps 1 and 
V must be equal to the sum of the diameters of 
steps 2 and 2'. When, however, an open belt is 
used, as is usually the case, the sum of the diam¬ 
eters of the steps at or near the middle of the cone 
will have to be somewhat greater than the sum of 
the diameters of those at or near the ends. 

What is generally considered to be the best 
method of determining the size of the various 
steps of cone pulleys is that given by Mr. C. A. 
Smith in the “Transactions of the American Society 
of Mechanical Engineers, Vol. X, page 269. Make 
the distance C, Fig. 180, equal to the distance 
between the centers of the shafts, and draw the 
circles A and B equal to the diameters of a known 
pair of steps on the cones. At a point midway 
between the shaft centers erect the perpendicular 
ab. Then, with a center on ab at a distance from 


242 


SELF-TAUGHT MECHANICAL DRAWING 


a equal to the length of C multiplied by 0.314, draw 
the arc c tangent to the belt line of the given pair 
of steps. The belt line of any other pair of steps 
will then be tangent to this arc. 

If the angle which the belt makes with the line 
of centers, de, exceeds 18 degrees, however, a 
slight modification of the above is made as follows: 
Draw a line tangent to the arc at c at an angle 
^ of 18 degrees with de; and with a center on ab, at 



Fig. 180.—Method of Laying out Cone Pulleys. 


a distance from a equal to the length C multiplied 
by 0.298 draw an arc tangent to this 18-degree 
line. 

All belt lines which make an angle with de 
greater than 18 degrees are made tangent to this 
new arc. 

The sizes of the steps so obtained may be verified 
by measuring the belt lengths of each pair. For 
this purpose a fine wire may be used, the wire 
being held in place by pins placed at close intervals 
on the outer half circumference of each pulley of 
the pair. 





















CHAPTER XV 


BOLTS, STUDS AND SCREWS 

Screws for clamping-work together are of three 
classes: through bolts, Fig. 181; studs, Fig. 182; 
cap screws. Fig. 183. In Fig. 181 the bolt is put 
entirely through both of the two pieces to be 



Fig. 181.—Through Bolt for 
Holding two Pieces to¬ 
gether. 



Clamping one Piece to 
another. 


clamped together, and a nut is put onto the 
threaded end. This is considered to be the best 
method on cast iron work, both as regards efficiency 
and cheapness, as there is no tappuig of any holes 

21 ;’, 














































244 SELF-TAUGHT MECHANICAL DRAWING 

in the cast iron. A tapped hole in cast iron is to 
be avoided, if possible, as, on account of the brittle 
nature of the material, the threads are liable to 
crumble or wear away easily. 

In many cases, however, it is not practicable to 
avoid tapping holes in cast iron, or questions of 
appearance may make the broad flange which is 
necessary when through bolts are used, undesirable. 
In such cases studs should be used. A stud consists 
of a piece of round stock threaded on both ends, 

and having a plain portion 
in the middle. The studs 
are screwed flrmly into the 
tapped holes, which should 
be deep enough to prevent 
the studs from bottoming 
in them, the studs instead 
binding or coming to a 
bearing at the end of the 
threaded portion. The 
loose piece is then put on 
over the studs, and is held 
in place by the nuts. By 
using studs, any further wear of the tapped hole 
is avoided, as, when removing the loose part, the 
nuts only are taken off, the studs being left in 
the body piece. 

When the material of the parts which are being 
clamped together is of such a nature that threads 
formed in it are not liable to crumble or to rapid 
wear, then cap screws. Fig. 183, may be used to 
advantage. They give a neat appearance to a 
piece of work, and the nut is entirely eliminated. 



Fig. 183.—Cap Screw used 
for Clamping Purposes. 
























BOLTS, STUDS AND SCREWS 


245 


United States Standard Screw Thread.—The 
most commonly used of all screw threads is the 
United States standard thread. A section, indicat¬ 
ing the form of this thread, is shown in Fig. 184. 
The thread is not sharp neither at the top nor at 
the bottom, but is provided with a flat at both of 
these points, the width of the flat being one-eighth 
of the pitch of the thread. The sides of the thread 


X 

o 



Fig. 184.—Form of the United States Standard Thread. 


form an angle of 60 degrees with each other. The 
“pitch’' and the “number of threads per inch” 
should not be confused. The pitch is the distance 
from the top of one thread to the top of the next. 
If the number of threads is 8 per inch, then the 
pitch would be J inch; and the flat on the, top of 
a United States standard thread, which, as men¬ 
tioned, is one-eighth of the pitch, would be 1-64 
inch. If the number of threads per inch is known, 
the pitch may be found by dividing 1 by the num¬ 
ber of threads per inch, or 


Pitch 


_ 1 _ 

No. of threads per inch. 


If, again, the pitch is known and the number of 
threads per inch required, then 


No. of threads per inch 


__L_ 

Pitch. 







246 SELF-TAUGHT MECHANICAL DRAWING 


U. S. STANDARD SCREW THREADS. 


Bolts and Threads 


^ o 


5 

T6 

f 

a 

4 

i 

1 

U 

n 

n 

IV 

If 

n 

u 

2 

21- 

2V 

2| 

3 
31 
3V 
3| 

4 

4i 

4V 

4| 

5 

5| 

6 


(4 

V 

a 

CQ .a 

a 

^ I—^ 
t.1 


20 
18 
16 
14 
13 
12 
11 ,() 


+5 

o 

Pi S 

r-'H 

C 

Cj 

P 




10 

9 

8 

“f 

/ 

7 

6 

6 

5V 

5 

5 

4V 

4“ 

4 


0 
0 
0 
0 
1 
1 
1 
1 
1 
1 
1 
1 
2 
2 
3V2 
3V2 


.185 0, 
,240|0, 
.294'0, 
344 0. 
400 0, 
454 0. 
.507 0, 
.620,0. 
.731 0, 
.S3r,o, 
.940,0. 
.035 0. 

.i6o;o. 

.234 0. 


0062 

0074, 

0078: 

0089 

0096 

0104 

0113 

0125 

013S 

0153 

0178 

0178 

021)8 


491 0, 
6160 . 


9620 
176 0 


3V 
3 
3 

2| 
2| 
2|4 


2V 

2i 

2i 

2 | 

2i 


428 
629 
879 
100 
317 
567 
798 
028 
256 
480 
730 
953 
203 0 
4230 


o 


c3J 


0250 
0250; 
0277j 
0277: 
0312 
03121 
0357 
0357 
0384 
041311 
0413,12 
0435 14 


0454 

0476 


15 

17 


0500 19 


0500 

0526 

0526 


21 

23 

25 


. 785 
.994 
.227 
. 485 
.737 
.074 
. 405 
.731 
.142 
.976 
.909 
.940 
.069 
.296 
.621 
.045 
.566 
.18611 
.90412 
.721114 
.63515 
.648 17 


8^ 

>- 

0 ) vw 
= 


.049 0 
.977 0 
.110 0 
.1.50 0 
.196 0 
.249; 0 
.3071 0 
.442i 0 
.601 


Hex. Nuts and 
Heads. 


to • 

inx. 
P M 

^ CO 

;-'cS 


.027 
.045 
.068 
,093 
, 126 

, 162 
*>'.) 

, ..-V.W, i , 

,3(:2|U 
A2.)\W^, 
■ 550| 11 
694 Uj) 


1 

2 

If* 
3 2 
1 1 

1 tj 
'1^1 

2 2 

I- 

3 2 

1-i- 


893 

057 


0555 28 


758 

967 

274 


19 

21 

23 


515 

743 

051 

302 

023 

719 

620 

428 

510 

548 

641 

963 

32. 

753 

226 

763 

572 

267 

262 

098 


2 

2t^ 

2| 

2f^ 

21 

2n 

3V 

H- 

4| 

5 

5| 

51 

H 

6V 

6t 

7 {- 

7i 

8 

8f 

9i 


to 

u to 

3"S 

•+^ ► 

cS 


1.7 
H 2 
a 
H 

2 3 

3 2 

1 3 

T6 

2 it 
3'2 

1 

iiV 

If 

lr\ 

l4^ 

2V 

2-^- 

9 I 
•> 

i2U 

2^ 

3tV 

3t^^ 

‘RLl 

6 

4J1 

4_f* 

415 

5-5- 

^16 

5H 

6* 

61 3 
Oi 6 

7 3 

< 

7_f> 

‘ TG' 

7it 

8t\ 

81^ 


9 


T'S^ 


CO 

O 

3S 

CO O 

c.; 

< 


11 
T .. 

5 1 

6 4 
2 it 
.3 2 


H 

Wi 

1 - ‘ 

2/. 

2rV 

2VI 

93 

■“f 

2-' 
■“3 2 

8 3 

•413 

*4^ tr 

3f“ 

4.A 

4 9 Q 

5f 


w 

to • 
(I>23 
C fcf' 
-iii ? 
« C 

HP 

H 


to • 


jr f- 


1 

4 

5 
T 1 


i 

_9_ 
4 ; 


a 

4 

i 

1 

u 

li 
If 
LV 
If 
If 
If 
2 

21 - 
9 I 

21 
3 

3f 
3V- 
-o 3| 

7tV41- 


TiT 

7|V 
81 ■! 

811 

Q 9 

9ii 

10/^ 

101-1 


3 

1 iT 
1 
4 

_5_ 
1 G 
3 
?■ 
-7 
1. 


it 

Tg 
11 
1 6 
13 
1 >1 
15 
1 G 

lr% 
1/^ 
-*• 1 G 

lU 

IB 

1 15 
■'■16 
2-3^ 
tJ 


4V 

4| 

5 

5i 

5^ 

5| 

6 


9-7 

2B 

2B 

3tV 

3nT 

3B 

3B 

4t% 

4tV 

4if 

4B 

5f\ 

5H 

515 


Square 
Nut and 
Head. 


e 

<u 

c feo 

'''' 3 
fx- Q 


< 


4 -I 
114 
U 3 
G 4 
G 3 
G 4 

■•■GT 
1 1 5 
■••Gl 


■ G 4 

If 
iM 
2,8 
9 1 y 

■^G 1 

2-^- 

•^1 G 

9 5 3 
^Gl 

3, ^ 
3B 
3| 

Q5 7 

4 

4li 

4. H 

r:3i 

Off-f 

6 

6ff 

7t(} 

7ff 

8f 

sn 

9t\ 

9i 

101 

1011 

nil 

Ilf 

12f 

12H 






















































BOLTS, STUDS AND SCREWS 247 


For example, assume that the pitch is 0.0625 
inch. Then 


No. of threads per inch == 


1 

0.0625 


= 16. 


The accompanying table of United States stand¬ 
ard screw threads gives the standard number of 
threads per inch, corresponding to given diame¬ 
ters, the diameter at the root of the thread, the 
width of the flat at the top and bottom of the 
thread, the area of the full bolt body, and the 
area at the bottom of the thread. These dimen¬ 
sions are, of course, always the same with all 
manufacturers. As regards the sizes for hexagon 
nuts and heads, and square nuts and heads also 
given in the table, it may be said that all makers 
do not conform strictly to the sizes as given. The 
catalog of one large bolt manufacturing concern, 
which is at hand, gives the width across flats of 
finished bolt heads and nuts the same as the rough 
sizes given in the table, which, it will be seen, are 
founded on the rule that the width across the flats 
of the heads and nuts should equal one and one- 
half times the diameter of the body of the bolt, 
plus one-eighth of an inch. It will also be noticed 
that the thickness of the head or nut is the same 
as the diameter of the body of the bolt. 

With cap screws, although the length of the head 
is made the same as for bolts, or equal to the di¬ 
ameter of the bolt body, the diameter of the head, 
and the distance across flats, is made different as 
shown in table on the following page: 



248 SELF-TAUGHT MECHANICAL DRAWING 


CAP SCREW SIZES. 
{From catalog of Boston Bolt Co.) 


Size of Screw 

1 

I 

5 

TS- 

3 

8 

7 

1 6 

1 

2 

9 

5 

'8 

3 

¥ 

7 

8 

1 

Width Across Flats 
Hex. Head 

Tf 

1 

2 

9 

16 

5 

8 

3 

? 

1 3 

1 6 

7 

■g- 

1 


n 

Width Across Flats 
Square Head 

3 

8 

7 

1 

2 

9 

1 6 

5 

8 

1 1 

1 6 

3 

¥ 

7 

8 

n 



Check or Lock Nuts.—When a bolt is subjected 
to constant vibrations there is a tendency for the 
nut to work loose. To overcome this tendency it 
is customary to employ a second nut, called a check 
or lock nut, which is screwed down upon the first 
one as shov/n in Fig. 185. When the first nut is 

screwed down to a bear¬ 
ing, the upper surfaces of 
its thread are in contact 
with the under surfaces 
of the bolt thread. When 
the check nut is screwed 
down, however, it forces 
the first nut down so that 
the under surfaces of its 
thread come into contact 
with the upper thread sur¬ 
faces of the bolt. This 
means that the check nut has to bear the entire load. 
When, therefore, the two nuts are of unequal 
thickness, as is frequently the case, the thick 
nut should be on the outside. 

Bolts to Withstand Shock—When a bolt which 
is subjected to shocks fails, it breaks, of course, 



Fig. 185.—Correct Arrange¬ 
ment when Using Check 
or Lock Nut. 











































BOLTS, STUDS AND SCREWS 


249 


at the part having the least cross sectional area, 
that is, at the bottom of the thread. If now the 
body of the bolt be reduced so that its cross section 
is of the same area as the area at the bottom of 
the thread, a slight element of elasticity is intro¬ 
duced, and the bolt is likely to yield somewhat 
instead of breaking. This is considered very im¬ 
portant in some classes of work. The reduction of 
area may be accomplished by turning down the 
body of the bolt, or, according to some authorities, 
the same object is attained by removing stock from 
the inside by drilling into the bolt from the head 
end. 

Either method, it is stated, gives the same degree 
of elasticity to the bolt, but as the drilling method 
takes the stock from the center, the bolt is left 
stiffer to resist bending or twisting than when the 
stock is taken off the outside by turning. 

Wrench Action.—When bolts or any form of 
screws are used to hold machine parts together, 
they must be strong enough not only to withstand 
the strain which is put upon them by the operation 
of the machine, but also to withstand the strain 
which is put upon them by the wrench in setting 
or screwing them up. In the case of a cylinder 
head, for instance, the strain upon the bolts due to 
the working of the engine will be the exposed area 
of the head, multiplied by the pressure per square 
inch. This divided by the number of bolts used 
will give the proportional part of this strain which 
each bolt must sustain. But in order to insure a 
tight joint, it is necessary that the bolts be not 
merely brought up to a bearing, but that they be 


250 SELF-TAUGHT MECHANICAL DRAWING 

set up hard enough so as to press the cylinder and 
cylinder head surfaces firmly together. The force 
which the wrench exerts in doing this work will 
be equal to the circumference of the circle through 
which the hand moves in turning the wrench 
through one revolution, multiplied by the force in 
pounds exerted at the handle, and this product 
divided by the distance through which the nut 
advances in one revolution, that is, by the lead of 
the screw. This theoretical result is, of course, 
modified by the friction between the nut and the 
bolt, and between the nut and washer. In addition 
to this direct strain, there is also a twisting strain 
in the bolt, caused by the friction between the bolt 
and nut. 

To insure the bolts being sufficiently strong to 
resist these various forces, it is customary to make 
them somewhat more than double the strength 
that would be necessary to enable them to safely 
resist the pressure of the steam or other fluid in 
the cylinder; that is, they are made about double 
strength to enable them to resist the direct strain 
of the wrench action, and then this amount is in¬ 
creased about 15 or 20 per cent, to allow for the 
twisting action of the wrench. Allowing that a 
factor of safety of 4 would be sufficient to allow 
for the steam pressure only, a factor of safety of 
not less than about 9 or 10 would therefore be used 
to provide for the added strain on the bolt due to 
the wrench action. In the case of small bolts, 
where the workman might set them up much harder 
than is really necessary, a factor of safety of about 
15 may be used. 


BOLTS, STUDS AND SCREWS 


251 


The distance apart which bolts can be spaced 
without danger of leakage is given by Prof. A. W. 
Smith as between 4 or 5 times the thickness of the 
cylinder flange for pressures between 100 and 150 
pounds per square inch. 

In the case of bolts which are not under strain 
as a result of the wrench action, as in the case of 



Fig. 186. —Example of Thread 
not under Stress due to 
Wrench Action. 



Fig. 187. —Square Threaded 
Screw, such as is Generally 
used for Power Transmis¬ 
sion. 


thh hook bolt shown in Fig. 186, a factor of safety 
as low as 4 might be properly used, if the load is 
steady. 

Assuming that the material of which the bolts 
are made has an ultimate strength of 40,000 to 
60,000 pounds per square inch, the factors of 
safety previously indicated would give allowable 
working stresses of from 4000 to 15,000 pounds per 
square inch. 













252 SELF-TAUGHT MECHANICAL DRAWING 


Screws for Power Transmission.—In Fig. 187 is 
shown a square threaded screw such as is generally 
used for power transmission. In such a screw the 
depth of the thread is made one-half of the pitch. 
The size of the body of the screw, assuming that 
the work which the screwds doing brings a ten- 
sional stress on the screw, will be determined by 
the tensile strength of the material of which it is 
made and the factor of safety which is used. As 
a screw which is used for power transmission is 
subjected to constant wear when in use, the ques¬ 
tion of the proper amount of bearing surface in the 
threads of the nut is of first importance, in order 
that it may not wear out too rapidly. The area of 
the thread surface in the nut on which the pressure 
bears will be equal to the difference in area of a 
circle of a diameter equal to the outside diameter 
of the screw, and one of a diameter equal to the 
diameter at the root of the thread of the screw, 
multiplied by the number of threads; or, letting D 
represent the outside diameter of the screw, and d 
represent the diameter of the body, the area will be: 

(D‘^ — d^) X 0.7854 X No. of threads in the nut. 

The allowable pressure per square inch of working 
surface will vary with the nature of the service 
required, whether fast or slow, and also with the 
lubrication, and with the material used. Where 
the speed is slow, say not over 50 feet per minute, 
and the service is infrequent, as in lifting screws, 
a pressure of 2500 pounds for iron or 3000 pounds 
for steel is allowable, while for more constant 
service some authorities limit the pressure to 
about 1000 pounds per square inch even when the 


BOLTS, STUDS AND SCREWS 


253 


lubrication is good. For high speeds a pressure of 
about 200 or 250 pounds is considered to be as 
much as should be allowed. 

For a screw which, fitting loosely in a well lubri¬ 
cated nut, is to sustain a load without danger of 
running down of itself, the pitch of the screw 
should not, according to Professor Smith, be greater 
than about one-tenth of its circumference. 

Efficiency of Screws.—A square-threaded screw 
has a greater efficiency than a V-threaded one, as 
the sloping sides of the V-thread cause an increase 
of friction. Square threads are therefore preferable 
for power transmission. Experiments show that 
in the case of bolts used for fastenings, the friction 
of the nut on the bolt and washer may absorb 90 
per cent, of the power applied to the wrench, 
leaving o-nly 10 per cent, for producing direct com¬ 
pression. For square-threaded screws an efficiency 
of about 50 per cent, is considered fair, if the 
screws are well lubricated. 

Acme Standard Thread.—While the square thread 
gives the greatest efficiency in a screw it is not as 
strong as one having sloping sides. Fig. 188 shows 
a section of a screw thread called the Acme or 29- 
degree thread, .which is often used for replacing 
the square thread for many purposes, such as in 
screws for screw presses, valve stems, and the 
like. The use of such a screw permits the employ¬ 
ment of a split nut, when such construction is 
desirable, which would not be practicable with a 
perfectly square thread, and for this reason, as 
well as for the reason that it can be cut with 
greater ease than the square thread, it has of late 


254 SELF-TAUGHT MECHANICAL DRAWING 

become widely used. In the Acme standard thread 
system the threads on the screw and in the nut are 
not exactly alike. A clearance of 0.010 inch is 
provided at the top and at the bottom of the thread, 
so that if the screw is 1 inch in diameter, for 
example, then the largest diameter of the thread 
in the nut would be 1.020 inch. If the root diam¬ 
eter of the same screw were 0.900 inch, then 
the smallest diameter of the thread in the nut 
would be 0.920 inch. The sides of the threads, 
however, fit perfectly. 

The depth of an Acme thread equals one-half the 
pitch of the thread plus 0.010 inch. The width 



Fig. 188.—Shape of Acme Screw Thread. 


of the flat at the top of the screw thread equals 
0.3707 times the pitch; and the width of the flat 
at the bottom of the thread equals 0.3707 times the 
pitch minus 0.0052 inch. 

Miscellaneous Screw Thread Systems.—Besides 
the screw thread systems already mentioned, a 
great many other systems are in more or less 
common use. Leading among these is the sharp 
V-thread, which, previous to the introduction of the 
United States standard thread, was the most com¬ 
monly used thread in this country. This thread 
is, theoretically at least, sharp at both the top and 
the bottom of the thread, the angle between the 









BOLTS, STUDS AND SCREWS 255 

sides of the thread being the same as in the United 
States standard system, or 60 degrees. In ordinary 
practice, however, a small flat is provided on the 
top of the thread, because it would be almost impos¬ 
sible to commercially produce the thread otherwise; 
and even if the thread could be produced, the sharp 
edge at the top would rapidly wear away. The 
sharp V-thread is being more and more forced 
out of use by the United States standard thread, 
although it must be admitted that it will probably 
long hold its own in steam fitting work, because of 
being especially adapted for making steam-tight 
joints. It answers this purpose probably better 
than any of the other common forms of threads. 

The Whitworth standard thread is not used to a 
very great extent in the United States, but it is the 
recognized standard thread in Great Britain. In 
this form of thread the sides of the thread form 
an angle of 55 degrees with each other, and the 
tops and bottoms of the threads are rounded to a 
radius equal to 0.137 times the pitch. This round¬ 
ing of the thread at the top provides for a thread 
which does not wear rapidly, and screws and nuts 
made according to this thread system will work 
well together in continuous heavy service for a 
longer period than would screws and nuts with any 
of the other standard thread forms. The fact that 
the threads are rounded in the bottom is advan¬ 
tageous on account of the elimination of sharp 
corners from which fractures may start. The main 
disadvantage of the thread, and the reason why 
the United States standard thread was adopted in 
this country in preference to the Whitworth stand- 


256 SELF-TAUGHT MECHANICAL DRAWING 

ard, which is the older of the two, is to be found 
in the fact that it is more difficult to produce than 
a 60-degree thread with flat top and bottom. The 
Whitworth form of thread is used in this country 
mostly on special work and on stay-bolts for loco¬ 
motive boilers. 

A thread perhaps more commonly used than any 
of the others, with the exception of the United 
States standard thread, is the Briggs standard 
pipe thread, which is used, as the name indicates, 
for pipe fittings. This thread is similar to the 
sharp V-thread, having an angle of 60 degrees 
between the sides, and nearly sharp top and bottom; 
instead of being exactly sharp at the top and bot¬ 
tom, however, it is slightly rounded off at these 
points. The difficulty of producing these slightly 
rounded surfaces has brought about a modification, 
at least in the United States, so that a small flat is 
made at the top, and the thread made to a sharp 
point at the bottom. It appears that a thread cut 
with these modifications serves its purpose equally 
as well as a thread cut according to the original 
thread form. 

Besides these systems, there are the metric screw 
thread systems. These use the same form of thread 
as the United States standard system, but the 
thread diameters and the corresponding pitches 

are, of course, made according to the metric system 
of measurement. 

Other Commercial Forms of Screws.—Set-screws, 
shown in Fig. 189, are usually made with square 
heads, and have either round or cup-shaped points, 
and are generally case hardened. They are used 


BOLTS, STUDS AND SCREWS 


257 


for such work as -fastening pulleys onto shafts, 
etc. Some set-screws are made headless, and are 
slotted for use with a screw-driver in places where 
it is undesirable that the 
screw projects beyond the 
work. 

The term machine screws 
covers a number of styles of 
small screws made for use 
with a screw-driver. Fig. 

190 shows the principal styles. 

Machine screw sizes are usu¬ 
ally designated by numbers, 
the size and the number of 
threads per inch being usually given together, with 
a “dash^^ between; thus a 10—24 screw would be a 
number 10 screw with 24 threads per inch. There 
are two standard systems for machine screw 


Fig. 189.—Forms of 
Set-screws. 



ROUND 

HEAD 


FILLISTER 

HEAD 



HEAD 


Fig. 190.—Forms of Machine Screws. 


threads, the old, which until recently was the only 
system, and the new, which was approved in 1908 
by the American Society of Mechanical Engineers. 
The standard thread form of the old system was 
the sharp V-thread, with a liberal but arbitrarily 












































































258 


SELF-TAUGHT MECHANICAL DRAWING 


selected flat on the top. The- basic thread form 
of the new system is that of the United States 
standard thread. 

The accompanying tables give the numbers and 
corresponding diameters and number of threads 
per inch of the old as well as the new system for 
machine screw threads. 

MACHINE SCREW THREADS, OLD SYSTEM. 


Number. 

Diameter. 

Threads 
per inch. 

Number. 

Diameter. 

Threads 
per inch. 

1 

0.071 

64 

12 

0.221 

24 

1.V 

0.081 

56 

13 

0.234 

22 

2 

0.089 

56 

14 

0.246 

20 

3 

0.101 

48 

15 

0.261 

20 

4 

0.113 

36 

16 

0.272 

18 

5 

0.125 

36 

18 

0.298 

18 

6 

0.141 

32 

20 

0.325 

16 

7 

0.154 

32 

22 

0.350 

16 

8 

0.166 

32 

24 

0.378 

16 

9 

0.180 

30 

26 

0.404 

. 16 

10 

0.194 

24 

28 

0.430 

14 

11 

0.206 

24 

30 

0.456 

14 


MACHINE SCREW THREADS, NEW SYSTEM. 


Number. 

Diameter. 

Threads 
per inch. 

! 

1 Number. 

j 

Diameter. 

Threads 
per inch. 

0 

0.060 

80 

12 

0.216 

28 

1 

0.073 

72 

14 

0.242 

24 

2 

0.086 

64 

16 

0.268 

22 

3 

0.099 

56 

18 

0.294 

20 

4 

0.112 

48 

20 

0.320 

20 

5 

0.125 

44 

22 

0.346 

18 

6 

0.138 

40 

24 

0.372 

16 

7 

0.151 

36 

26 

0.398 

16 

8 

0.164 

36 

28 

0.424 

14 

.9 

0.177 

32 

30 

0.450 

14 

10 

0.190 

30 

( 









































CHAPTER XVI 


COUPLINGS AND CLUTCHES 

A COUPLING is a device for connecting together 
the ends of two shafts or axles for the purpose 
of making a longer shaft, the term being usually 
limited to those devices which are intended for 
permanent fastening. The term clutch is used to 
designate a disengaging coupling. 

The simplest form of coupling consists simply of 
a sleeve or muff, made of a length about three 
times the diameter of the shaft, bored out to fit 
the shaft, and provided with a keyway its entire 
length; made to receive a tapering key. The ends 
of the shafting are, of course, also provided with 
keyways, and are inserted into the sleeve; then 
the key is driven in. In some couplings the sleeve 
is made tapering on the outside at both ends, and, 
being split, is clamped upon the shafts by means 
of rings or hollow conical sleeves which are driven 
onto the tapered ends, or drawn together by means 
of bolts. 

One of the most common forms of coupling is the 
flange coupling shown in Fig. 191. In this case a 
flanged hub is keyed to each of the shaft ends, and 
the flanges are then held together and prevented 
from turning relative to each other by bolts, as 
shown. In some cases the bolt heads and nuts are 

259 


260 SELF-TAUGHT MECHANICAL DRAWING 

provided with a guard by having the rim on the 
outer edge of the flange made deep as shown by 
the dotted lines on one side. This construction 
also allows the coupling to be used as a pulley, if 
necessary. In a coupling of this kind, the chief 
problem is to get the bolts of such size that their 
combined strength to resist the shearing action 
to which they are subjected equals the twisting 



strength of the shaft. Letting d represent the 
diameter of the shaft in inches, its internal resist¬ 
ance to twisting is given by the formula 

5.1 

in which T equals the internal resistance to twist¬ 
ing, or the twisting moment, and S the shearing 
strength per square inch of area in pounds. 

Regarding the shearing strength of materials 
Kent says: “The ultimate torsional shearing re¬ 
sistance is about the same as the direct shearing 
resistance, and may be taken at 20,000 to 25,000 
pounds per square inch for cast iron, 45,000 pounds 































COUPLINGS AND CLUTCHES ’ 261 


for wrought iron, and 50,000 to 150,000 pounds for 
steel according to its carbon and temper.’^ 

The torsional and direct shearing resistance being 
the same, this quantity may be neglected if the 
shaft and coupling bolts are of the same material, 
and 

AL 

5.1 

the internal resistance factor or torsion modulus 
of the shaft, should be equal to the product of the 
radius of the bolt circle of the coupling, the number 
of bolts used, and the area of each bolt. Or, letting 
a represent the area of each bolt, R the radius of 
the bolt circle of the coupling, and n the number 
of bolts used, we would have: 

a = (R X n). 


Example .—Required the size of the bolts for a 
flange coupling for a 2-inch shaft. The radius of 
the bolt circle is 3 inches, four bolts being used. 

Using the notation in the formula given, our 
known values are: 

d = 2 inches, 

R = S inches, 
n = 4: bolts. 


If we insert these values in the formula we have: 


a 


= (3 X 4) = A 12 = 0.13 square inches. 

5.1 o.i 


This area corresponds to a diameter of about tV 
of an inch. To allow for the strain on the bolt 
caused by the action oi the wrench, the next size 





262 SELF-TAUGHT MECHANICAL DRAWING 

larger bolt, at least, or a J inch bolt, will be se¬ 
lected. The capacity of the bolt to resist shear¬ 
ing will be considerably increased by having the 
corners of the holes at those faces of the flanges 
which come together, somewhat rounded. If this 
is not done, the action of the flanges on the bolts 
will be like that of a pair of sharp shears. Experi¬ 
ments have shown that with the corners rounded, 
the capacity of the bolt to resist shearing may be 

t 

ini'reased 12 per cent. 

If the shaft and bolts are of different materials 
then the modulus 

5.1 

should be multiplied by the shearing strength of 
the shaft in pounds per square inch and the product 



Fig. 192.—Clamp Coupling. 


R X % should be multiplied by the shearing strength 
of the bolts per square inch, before dividing in 
the formula to get the bolt area. 

In Fig. 192 is shown another form of coupling 
much used. It consists of two parts bolted together- 
over the joint in the shafting, a key and keyway 
being provided to prevent the slipping of the shafts. 





















































COUPLINGS AND CLUTCHES 2G3 

By having a thickness of heavy paper interposed 
between the two parts of the coupling when it is 
bored out, it may be made to clamp very tightly 
onto the shafts. 

With either form of coupling, the length is made 
such that each shaft end is held by the coupling 
by a length of about one and one-half times its 
diameter, as indicated in the engravings. 

Oldham’s Coupling.— Fig. 193 shows a form of 
coupling which may be used for shafts which are 



Fig. 193.—Oldham’s Coupling. 



parallel, but slightly out of line. In this coupling 
each shaft end has a flanged hub attached to it. 

Across the face of each flange is planed a single 
groove passing through its center. Interposed 
between the two flanges is a disk, shown at the 
right, having tongues on both faces at right angles 
to each other, to engage in the grooves in the 
flanges. 

Hooke’s Coupling or Universal Joint is used for 
connecting two shafts whose axes are not in line 
with each other, but merely intersect. The shafts 
A and B, and B and C, in Fig. 194, are thus con¬ 
nected by universal joints. If the shaft B is made 
telescoping, as is very often the case, a solid part 


























264 SELF-TAUGHT MECHANICAL DRAWING 

entering into and being keyed in a sleeve so as 
to prevent independent rotation, but yet permit 
a sliding action, then the two shafts A and C may 
move independently of each other within certain 
limits, the distance between their ends being 
capable of variation. The arrangement shown in 
Fig. 194 is used on various machine tools, notably 
on milling machines, flange drilling machines, etc. 
Many designs of flexible shafts are really only a 
combination of a great number of universal joints. 



Fig. 194.—Application of Universal Joints and Telescoping 

Shaft. 


When this coupling is employed for driving only 
one shaft at an angle with another, as if shaft 
A simply drove shaft B which, of course, is the 
fundamental type of universal coupling, then, if 
the driving shaft has a uniform motion, the driven 
shaft will have a variable motion, and so cannot be 
used in such cases where uniformity of motion of 
the driven shaft is necessary; but where there are 
three shafts, as shown in the illustration, A will 
impart a uniform motion to C provided the axes of 
A and C are parallel with each other, as shown; for 
if A, having a regular motion, imparts an irregular 
























COUPLINGS AND CLUTCHES 


265 


motion to B, then if B, with its irregular motion, 
is made the driver, it will impart a regular motion 
to A, and as C is parallel with A it will also impart 
a regular motion to C. 

This form of coupling does not work very well 
if the angle a is more than 45 degrees. 

Clutches are of two general classes, toothed 
clutches and friction clutches. An example of a 
toothed clutch is shown in Fig. 195. In this clutch 
the part at the left is fastened to its shaft; the 
part at the right is free to slide back and forth upon 



Fig. 195. —A simple Form of Toothed Clutch. 


its shaft, but is prevented from turning on the 
shaft by a key. The sliding motion for engaging 
or disengaging this part of the clutch is accom¬ 
plished by means of the forked lever and jointed 
ring, shown at the right, which latter engages in 
the groove A. Such a clutch, while giving a pos¬ 
itive drive, cannot, of course, be thrown in or out 
while the driving shaft is running at a high rate 
of speed. By having the back faces of the teeth 
beveled off as shown by the dotted lines, this diffi¬ 
culty is partly overcome, although the shock caused 
by the sudden engaging of the teeth still renders 


























266 SELF-TAUGHT MECHANICAL DRAWING 

the clutch unsuitable for operating at very high 
speed. To facilitate uncoupling, the driving faces 
may also be given an angle of about 10 or 12 
degrees. 

Friction Clutches are generally made in one of 
the two styles shown in Figs. 196 and 197. The 
power which a clutch of the type shown in Fig. 196 
will transmit, depends upon the power which is ap¬ 
plied to force the sliding part against the fixed part, 



and the efficiency of the frictional force between 
the rubbing surfaces. As to the efficiency of the 
clutch, therefore, much depends upon the nature of 
the engaging surfaces, whether metal comes in 
contact with metal, or whether one of the surfaces 
has a facing of leather or wood. The efficiency is, 
of course, much increased by either a leather or 
wood facing. Professor Smith gives the efficiency 
of these different surfaces as follows: Cast iron on 
cast iron, 10 to 15 per cent.; cast iron on leather. 
































COUPLINGS AND CLUTCHES 2G7 

20 to 30 per cent.; cast iron on wood, 20 to 50 per 
cent. 

The horse-power which such a clutch will trans¬ 
mit will be found by multiplying the velocity of the 
parts in contact, in feet per minute, taken at their 
mean diameter as indicated at D, by the force 
which is being applied at this diameter in the 
direction of revolution, and dividing this product 
by 33,000. The force which is acting at the diameter 
D to produce revolving motion is equal to the pres¬ 
sure which is being applied to force the two parts 
of the clutch together, multiplied by the coefficient 
of friction (as the frictional efficiency between the 
surfaces in contact, as given above, is called) of the 
materials which form the driving surfaces. 

Example .—What power will a clutch of the type 
shown in Fig. 196 transmit if running at a speed 
of 250 revolutions per minute? The diameter D is 
18 inches, and a pressure of 50 pounds is exerted 
to force the two clutch faces together. One of 
the clutch parts has a leather facing, and the 
coefficient of friction is 0.25. 

The general formula for finding the horse-power 
of a clutch of this type is: 

^ " 33,000 

in which H.P. = horse-power transmitted, 

D = mean diameter of friction sur¬ 
faces in feet, 

n = revolutions per minute, 

P = pressure between clutch surfaces 
in pounds, 

/ = coefficient of friction. 


268 SELF-TAUGHT MECHANICAL DRAWING 


The values to be inserted in the formula, which 
are given in this problem, are as follows: 

18 


D = 


12 


1.5 foot. 


n = 250 revolutions, 

P = 50 pounds, 

/ = 0.25. 

Inserting these values in the formula we have: 

1.5 X 3.1416 X 250 X 50 X 0.25 ^ 

=--= 0.45. 


The formula given may be transposed in various 
ways according to the requirements of the problem; 
if, for instance, it is desired to know what pressure 
must be applied to transmit a given horse-power, 
then: 


_RP. X_33,000_ 

D (in feet) X 3.1416 X n X f. 


If the pressure is known, and it is required to 
find what diameter the clutch must be made to 
transmit a given power, then: 


D (in feet) 


H.P. X 33,000 

3.1416 X n X P X f. 


If the pressure and diameter are both known, 
then the number of revolutions which the clutch 
must make per minute to transmit a given horse¬ 
power will be: 

_ _ H.P. X 33,0 00_ 

” D (in feet) X 3.1416 X P X /. 


It may be said that the capacity of the clutch 
to transmit power is independent of the area of the 








COUPLINGS AND CLUTCHES 


269 


friction surfaces; for, if the friction surface is 
increased the pressure which is applied to force 
the two parts of the clutch together is simply dis¬ 
tributed over a much greater area, giving a smaller 
pressure per square inch. The durability would be 
increased, but the horse-power capacity would re¬ 
main unchanged. 

The conical clutch shown in Fig. 197 may be 
made to run metal to metal, or the hollow part may 



Fig. 197.—Friction Cone Clutch. 


be made larger to allow of the insertion of wooden 
blocks. This would increase the efficiency, but at 
the expense of the durability. The principle of 
this form of clutch may be explained by referring 
to the diagrammatical sketch at the right of Fig. 
197, where the angle ACB represents the angle 
which the opposite sides of the clutch make with 
each other, the line DC representing the axis of 
the shaft. If now line hd of the small triangle 
abd be considered as representing the magnitude 




























270 SELF-TAUGHT MECHANICAL DRAWING 


of the force acting in the direction of the axis 
of the shaft to force the two parts of the clutch 
together, then if ah is at right angles to AC, ah 
will represent the resultant magnitude of the force 
acting on the face of the clutch at right angles to 
its surface, according to the principles explained 
in the chapter on the elements of mechanics. The 
efficiency of the clutch will therefore be as much 
greater than that of a flat-faced clutch as ah is 
greater than hd. The horse-power of such a 
clutch, using the same notation as before, would, 
therefore, be: 

up _ (in feet) X 3.1416 XnXPXf^ ah 
" MMO hd. 

But from the chapter on the solution of triangles 
we know that 

« 

=sine of angle had. 

Hence 

ah _ _J. 

hd sin had. 


But angle had equals angle x, the angle which 
the conical surface of the clutch makes with the 
axis of the shaft. 

Therefore 

ah _ 1 

hd sin X 


and our original formula takes the form: 




£L(inJ[eet) X 3.141 6 X n X P X f 
33,000 X sin x. 





COUPLINGS AND CLUTCHES 271 


Transposing this formula as before for the flat¬ 
faced clutch, gives us: 

p ^ H.P. X 33,000 X si n x 
D (in feet) X 3.1416 X nX~f. 


D (in feet) 


n 


^ X 33,00 0 X si n x 

3.1416 XnX PX/r 

H. P. X 33, 000 X sin x 


D (in feet) X 3.1416 y P X f. 


The sine of x may be taken from the tables of 
trigonometric functions previously given in the 
chapter on the solution of triangles, or it may be 
found by dividing the length hd (Fig. 197) by the 
length ah. 

The power necessary to force the two parts of 
the clutch together may be neglected, as the slip¬ 
ping which occurs as they are engaging allows 
them to come together with but little pressure 
beyond what is required for power transmission 
purposes. The angle which the face of the clutch 
makes with the shaft (the angle x in the diagram 
at the right in Fig. 197) should be such that the 
clutch does not grip too quickly when thrown into 
gear, nor require too much pujl to release. Making 
this angle between 7 and 12 degrees conform.s to 
the average given by different authorities. 








CHAPTER XVII 


SHAFTS, BELTS AND PULLEYS 

Shafts.—The twisting strength of a shaft, as 
stated in the preceding chapter, is given by the 
formula 

^ 5.1 

in which T = twisting moment, or force which 

acting at a distance of one inch 
from the center of the shaft would 
produce in it a torsional shearing 
stress of S pounds per square inch, 
d = diameter of shaft in inches, 

S = torsional shearing stress in pounds 
per square inch. 

Expressing this formula in words we may say 
that the cube of the diameter in inches multiplied 
by the torsional shearing stress, and this product 
divided by 5.1, gives the force which acting at a 
distance of one inch from the center of the shaft 
would produce in it the given torsional shearing 
stress. 

The twisting moment T equals, therefore, the 
force Fi , acting at a distance of one inch from 
the center of the shaft, times 1; it also equals any 
other force F exerting a twisting action on the 

272 



SHAFTS, BELTS AND PULLEYS 


273 


shaft multiplied by its distance from the center of 
the shaft. The formula given can hence be written 

/Ti d ^ yc s 

T = F X r = —— 

0.1 

in which F = any force acting at a distance r from 
the center of the shaft. 

Transposing this formula to obtain the distance 
from the center (r) at which a given force would 
have to act to set up a torsional shearing stress 
in the shaft, we would have: 

S 

^ 5.1 XF. 

The force which would be necessary to set up 
a stress in the shaft when acting at a given 
distance would be: 

d^_x_S 
^ 5.1 Xr. 


The diameter of shaft to resist a given force 
acting at a given distance would be: 



I F X r X 5 .1 
S 


The torsional shearing strength of ordinary 
shafting is about 45,000 pounds to the square inch, 
and of steel shafting from about 50,000 to 150,000 
pounds, according to its quality; these figures 
should be divided by five or six to give a safe 
working stress. 

The above formulas, however, are based on the 
assumption that the force acting is of a purely 







274 SELl^'-TAUGIIT MECHANICAL DRAWING 


twisting nature, as if a hand-wheel were put onto 
the end of the shaft, and the tendency to bend the 
shaft, caused by the pull of one hand, were counter¬ 
acted by the push of the other hand. In the case 
of a shaft actuated by a rocker arm, as sometimes 
occurs in machines, the tendency to bend the shaft 
caused by the push on the arm could be provided 
for by using a somewhat higher factor of safety. 
If the arm were placed at some distance from the 
bearing, however, the tendency to bend the shaft 
might be greater than the twisting effect. 

The methods of calculating the size of shafts for 
transmitting a given power, so as to take into 
account both the twisting and bending effects pro¬ 
duced by the pull of the belt are quite complicated, 
and the beginner will ordinarily find it best to use 
some of the empirical formulas for that purpose 
which are intended to take into account both of 
these effects. 

The following rules by Thurston are considered 
to afford ample margin for strength for shafts 
which are well supported against springing: 

To find the diameter of a cold rolled iron shaft to 
transmit a given horse-poicer, multiply the horse¬ 
power to he transmitted by 75, and divide the product 
by the number of revolutions per minute that the 
shaft is to make. The cube root of this quotient will 
he the diameter of the shaft. 

If the shaft is to be of turned iron, proceed as 
above, except that the horse-power to be trans¬ 
mitted is to be multiplied by 125 instead of 75. 

This rule is “for head shafts, supported by bear- 


SHAFTS, BELTS AND PULLEYS 275 

ings close to each side of the main pulley or gear, so 
as to wholly guard against transverse strain/^ If 
the main pulley is at a distance from the bearing, 
the size of the shaft will need to be increased, 
while for ordinary line shafting, with hangers 8 
feet apart, the size may be reduced, figures of 90 
for turned iron, and 55 for cold rolled iron shafting 
being substituted for those given in the rule; or, in 
the case of shafting for transmission only, without 
pulleys, figures of 62.5 for turned iron, and 35 for 
cold rolled iron are substituted. 

To find the horse-potver which a given shaft will 
transmity multiply the cube of its diameter by the 
number of revolutions per minute, and divide the 
product by 125 for turned iron, or by 75 for cold 
rolled iron. 

For line shafting substitute the figures given by 
90 and 55, respectively. 

The horse-power which is being transmitted is 
determined by multiplying the pull in pounds 
which the belt exerts (or the push which the teeth 
of the driving gear exert, if gears are used) by 
the diameter of the pulley in inches (or. the pitch 
diameter of the gear in inches) and multiplying 
this product, again, by the number of revolutions 
per minute of the shaft; then divide this product 
by 126,050, and the quotient gives the horse-power 
transmitted. 

Expressed as a formula this rule would be: 



276 SELF-TAUGHT MECHANICAL DRAWING 

in which P = pull on belt or push on gear teeth in 

pounds, 

D = diameter of pulley or pitch diameter 
of gear in inches, 

N = number of revolutions per minute of 
pulley or gear. 

Belts.—The theoretical horse-power which a belt 
will transmit is equal to the pull which the belt 
exerts in pounds, multiplied by its velocity in feet 
per minute, and this product divided by 33,000. 
The question then arises as to what is the allowable 
stress to be put upon a belt. 

A common rule of practice for ordinary belting 
is that for single thickness belts the horse-power 
transmitted equals the breadth of the belt in inches, 
multiplied by its velocity in feet per minute, this 
product being divided by 1,000. This rule assumes 
a belt pull of 33 pounds per inch of width. Many 
authorities, however, would allow a much higher 
tension. The higher the tension, however, the 
narrower the belt for a given horse-power, and 
the greater the stretch, the more frequent the 
necessity for relacing, and the shorter the life of 
the belt. 

Allowing 33 pounds tension per inch in width for 
the thinnest commercial single belt, and allowing 
the tensions for increased thicknesses given by a 
large belt manufacturing concern, would give the 
following formulas for the transmission capacities 
of given belts: 


SHAFTS, BELTS AND PULLEYS 


277 


Single belt, inch thick, H.P. 

Single belt, inch thick, H.P. 

Light double, inch thick, H.P. 

Heavy double, iiich thick, H.P. 

Heavy double, ii^ch thick, H.P. 

Heavy double, f inch thick, H.P. 

Heavy double, inch thick, H.P. 

In these formulas the breadth of the belt is 
understood to be in inches, and its velocity in feet 
per minute, the letters if.P. meaning horse-power. 
Transposing the above formulas to ascertain the 
breadth of belt required to transmit a given power, 
we would have: 


Breadth X velocity. 

1000 

Breadth X velocity. 

800 

Breadth X velocity. 

733 

Breadth X velocity. 
687 

Breadth X velocity. 

660 

Breadth X velocity. 

550 

Breadth X velocity. 
500 


Single belt, 
Single belt, i 
Light double, H 
Heavy double, 
Heavy double, 
Heavy double, | 


inch thick. Breadth = 


inch thick. Breadth = 


inch thick. Breadth == 


inch thick. Breadth =- 


inch thick. Breadth = 


H P. X 1000 
Velocity 

H. P. X 800 
Velocity 

H. P. X 733 
Velocity 

H. P. X 687 
Velocity 

H. P. X 660 


Velocity 


inch thick. Breadth =- 


H. P. X 550 

Velocity 


Heavy double, inch thick. Breadth = 


H P. X 500 

Velocity 

















278 SELF-TAUGHT MECHANICAL DRAWING 

These formulas are all for laced belts. A belt 
made endless by being lapped and cemented or 
riveted is considered to be nearly 50 per cent, 
stronger than a laced belt, and is thus capable of 
transmitting nearly 50 per cent, more power; or 
the breadth of an endless belt to transmit a given 
power would not need to be more than between 
two-thirds to three-quarters of the breadth of a 
laced belt. Metal fastenings are not considered to 
make as strong a belt as lacings. 

If the foregoing formulas had been made on the 
basis of an allowable stress of 45 pounds for each 
inch in width of a single belt, a figure which many 
consider perfectly safe for a belt in good condition, 
they would have shown the belts as being capable 
of transmitting one-third more power than at 33 
pounds stress per inch; to transmit a given power 
a belt would then need to be not more than three- 
quarters of the width. 

It will be seen from these formulas that the 
power transmitting capacity of a belt depends upon 
its breadth (a wide belt allowing an increased 
tension) or on its velocity. Increasing the width 
of the belt without increasing the tension to corre¬ 
spond would not give any increase of power trans¬ 
mitting capacity, as the given tension would simply 
be distributed over so much more pulley surface; 
but a tight belt means more side strain on shaft 
and journal. Therefore, according to Griffin, from 
the standpoint of efficiency, lise a narrow belt under 
low tension at as high a speed as possible. The de¬ 
sired high speed is, of course, secured by simply 
putting on large pulleys. 


SHAFTS, BELTS AND PULLEYS 


279 


Speed of Belting.—The most economical speed 
is somewhere between 4000 and 5000 feet per 
minute. Above these values the life of the belt is 
shortened; '‘flapping,“chasing,’^ and centrifugal 
force also cause considerable loss of power at higher 
speeds. The limit of speed with cast iron pulleys 
is flxed at the safe limit for the bursting of the 
rim, which may be taken at one mile surface speed 
per minute. 

The formulas given for the horse-power trans¬ 
mitted assume that the belt is in contact with just 
one-half of the pulley; or, in other words, that the 
arc of contact is 180 degrees. If the arc of contact 
is increased, as it might be in the case of a crossed 
belt, until it becomes 240 degrees, or two-thirds of 
the circumference of the pulley, it is stated that 
the adhesion of the belt to the pulley, and conse¬ 
quently the efficiency of the belt, will be increased 
50 per cent. If, on the other hand, the arc of con¬ 
tact should be reduced to 120 degrees, or one-third 
of the circumference of the pulley, as might be the 
case with open belts where the shafts were near 
together, and the pulleys were very unequal in 
size, the efficiency is stated to be only 60 per cent, 
of what the formulas would show; if the arc of con¬ 
tact should be reduced to 90 degrees, the efficiency 
is stated to be only 30 per cent. 

From these percentages one can form a .fairly 
good idea of what percentage to allow for varying 
arcs of contact. In most cases, however, it will 
probably be correct enough to assume the arc of 
contact to be 180 degrees. 

In all cases of open horizontal belting the lower 


280 SELF-TAUGHT MECHANICAL DRAWING 

run of the belt should be made the working part, 
so that the sag of the upper run will increase the 
arc of contact. 

In the location of shafts that are to be connected 
with each other by belts, care should be taken* to 
secure them at a proper distance from one another. 
It is not easy to give a definite rule what this dis¬ 
tance should be. Some authorities give this rule: 
Let the distance between the shafts be ten times the 
diameter of the smaller pulley; but while this is 
correct for some cases, there are many other cases 
in which it is not correct. Circumstances generally 
have much to do with the arrangement; and the 
engineer or machinist must use his judgment, mak¬ 
ing all things conform, as far as may be, to general 
principles. The distance should be such as to allow 
a gentle sag to the belt when in motion. The Page 
Belting Co. states that if too great a distance is 
attempted, the weight of the belt will produce a 
very heavy sag, drawing so hard upon the shafts 
as to produce considerable friction in the bearings, 
while at the same time the belt will have an un¬ 
steady, flapping motion, which will destroy both 
the belt and the machinery. 

As belts increase in width they should be made 
thicker. It is advisable to use double belts on 
pulleys 12 inches in diameter and larger. If thin 
belts are used at very high speed, or if wide belts 
are thin, they almost invariably run in waves on 
the slack side, or “travel'' from side to side of the 
pulleys, especially if the load changes suddenly. 
This waving and snapping that occurs as the belts 
straighten out, wears the belts very fast, and 


SHAFTS, BELTS AND PULLEYS 281 

frequently causes the splices to part in a very short 
time, all of which is avoided by the employment of 
suitable thickness in the belts. The Page Belting 
Co. states that driving pulleys on which are to be 
run shifting belts should have a perfectly flat sur¬ 
face. All other pulleys should have a convexity in 
the proportion of about t\ of an inch to one foot 
in width. The pulleys should be a little wider than 
the belt required for the work. 

Pulley Sizes.—The sizes of pulleys to give a re¬ 
quired speed, or the speed which will be obtained 
with given pulleys may be readily found from the 
fact that the product of the speed of the driving 
shaft, in revolutions per minute, and the diameters 
of all driving pulleys, on the main and on counter¬ 
shafts, multiplied together, will be equal to the 
product of the diameters ^of all driven pulleys and 
the speed of the last driven shaft, in revolutions 
per minute, multiplied together; so that if the size 
of one driven pulley, for instance, is required, its 
size may be found by dividing the product of the 
speed of the driving shaft and all driving pulleys 
multiplied together, by the product of speed of the 
final driven shaft and the diameters of such driven 
pulleys as are given, multiplied together. The re¬ 
sult will be the required pulley size. 

Example .—A shaft making 200 revolutions per 
minute has mounted on it a pulley 18 inches in 
diameter which belts onto a 6-inch pulley on a 
countershaft. The countershaft has mounted on it 
a 20-inch pulley which belts to a pulley on the 
spindle of a machine which is to make 3000 revolu¬ 
tions per minute. What size pulley will be required 
on the spindle. 


282 SELF-TAUGHT MECHANICAL DRAWING 


Placing the speed of the driving shaft, and the 
sizes of all driving pulleys on one side of a vertical 
line, for convenience sake, and the sizes of all 
driven pulleys and the speed of the last driven 
shaft (or spindle) on the other side, and letting x 
represent the required size we would have: 


Speed of shaft = 200 

Pulley on shaft = 18 

Driving pulley on 

countershaft = 20 


{ 6 Driven pulley on coun¬ 

tershaft. 

X Required size of pulley 

on spindle. 

3000 = Speed of spindle. 


Then 200 Xl8 X 20 = 6X x X 3000 

_ 200 X 18 X 20 _ _72,000 _ . 

* 6 X 3000 18,000 

The diameter of the pulley on the spindle would 
therefore have to be 4 inches. If this size had 
been given, and the speed of the spindle had been 
required, x might have been taken to represent the 
required speed, when the same process would have 
given the desired information. 

Twisted and Unusual Cases of Belting. — It 
frequently happens that, in transmitting power, 
conditions present themselves in which ordinary 
straight belting, either open or crossed, will not 
serve the purpose, and recourse must be had to 
some form of twisted belting, either quarter turn 
belting or belting guided by idler pulleys. In the 
following are given some of the principal con¬ 
ditions. 

Fig. 198 shows a quarter turn belt, by which 
power can be transmitted from one shaft to another 
at right angles to it. The condition necessary for 



SHAFTS, BELTS AND PULLEYS 


283 


the successful working of this arrangement is that 
the middle of the face of the pulley toward which 





Fig. 198.— Arrangement of Fig. 199.— Another Arrange- 

Pulleys for Quarter-Turn ment for Transmitting 

Belt. Power between Shafts at 

Right Angles. 

the belt is advancing shall be in line with the edge 
of the pulley that the belt is leaving. An exami- 

































































284 SELF-TAUGHT MECHANICAL DRAWING 

nation of both the plan and elevation views will 
make this clear. 

While this is the simplest arrangement for this 
purpose, it has several drawbacks. The edgewise 
stress on the belt as it is leaving either pulley is 
very severe on the belt. It also causes a consider¬ 
able loss of contact with the pulley face, with 
corresponding loss of power transmission capacity. 
The edgewise stress also makes it necessary, if 
durability is to be considered, to have the belt 
relatively narrow. Incidentally, also, any reversal 
of the motion will cause the belt to immediately 
run off the pulleys. 

Fig. 199 shows another arrangement for trans¬ 
mitting power from one shaft to another at right 
angles to it, which overcomes all of the objections 
mentioned to the arrangement shown in Fig. 198, 
but at the expense of a double length belt and an 
extra pair of pulleys. 

As shown in the illustration, A and B are tight 
pulleys, and C and B are loose pulleys. The belt, 
as it leaves the tight pulley A, passes down under 
the loose pulley D, up over the loose pulley C, down 
under the tight pulley B, and then up over the 
tight pulley A, making a complete circuit. The 
loose pulleys, it will be seen, revolve in an opposite 
direction to the shafts on which they are mounted. 

Fig. 200 shows an arrangement by which, by 
employing loose guide pulleys, power may be trans¬ 
mitted from one shaft to another so close to it as 
to prohibit direct belting. If the main pulleys are 
of the same size, and their shafts are in the same 
plane, the guide pulleys may be mounted on a 


SHAFTS, BELTS AND PULLEYS 285 

single straight shaft at right angles to a plane 
passing through the axes of the shafts on which 
the main pulleys are mounted. If, however, the 
main pulleys are of unequal size, as shown in the 
illustration, the guide pulleys will have to be in¬ 
clined to such an angle that the center of the face 



Fig. 200.—Arrangement of Belt Transmission Using 
Loose Guide Pulleys. 


of the pulley toward which the belt is advancing 
shall be in line with the edge of the pulley that 
the belt is leaving, the same as in the case of the 
quarter turn belt shown in Fig. 198. 

It is not necessary that the shafts on which the 
main pulleys are mounted be in the same plane; 
their direction may be such that their relation to 




















286 SELF-TAUGHT MECHANICAL DRAWING 


each other is similar to that of those shown in 
Fig. 198, or at any intermediate angle. 

Again, if they are in the same plane, it is not 



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necessary that they should be parallel with each 
other; they may be at any angle with each other. 

Fig. 201 shows a case which is a modification of 
Fig. 200, The main shafts are at right angles to 


































SHAFTS, BELTS AND PULLEYS 287 

each other. The main pulleys, being of the same 
size, permit the guide pulleys to be mounted on 
a single shaft. This arrangement is a common 
method of transmitting power around a corner. 

Fig. 202 shows a case where the direction of the 
shafts with regard to each other is the same as in 



in an Adjustable Frame. 


Fig. 198, but where shop conditions are such that 
it is not practicable to bring the lower shaft under 
the upper one to permit of belting by either of the 
methods shown in Figs. 198 or 199. The guide 
pulleys are, therefore, mounted on a frame which 
can be raised or lowered in guides by means of an 
adjusting screw, permitting of an easy adjustment 
of the belt tension. 



















288 SELF-TAUGHT MECHANICAL DRAWING 


Fig. 203 shows a case which is similar to Fig. 
200 in that it permits the belting together of shafts 

which are at angle to each 
other, but accomplishes this 
result by the use of only one 
guide pulley. The shafts, 
though at an angle to each 
other, are in the same plane. 
This, however, is not neces¬ 
sarily so. The shafts may 
be twisted around until they 
are at right angles to each 
other, as in Fig. 198. As 
shown in Fig. 200, the belt 
may be run in either direc¬ 
tion as long as the shafts are 
in the same plane; but as 
shown in Fig. 203, it is nec¬ 
essary that the belt should 
be run in the direction in¬ 
dicated by the arrows. 

An examination of the en¬ 
gravings will show that the 
condition necessary for the 
proper working of guide 
pulleys is that the shaft on 
which the guide pulley is mounted shall be at right 
angles to a line drawn from the edge of the pulley 
that the belt is leaving in its advance toward the 
guide pulley, to the middle of the guide pulley 
face. 



Fig. 203.—An Arrange¬ 
ment in Which but One 
Guide Pulley is Used. 











CHAPTER XVIII 

FLY-WHEELS FOR PRESSES, PUNCHES, ETC. 

In a great many different classes of machinery, 
the work that the machine performs is of a variable 
or intermittent nature, being done, in the case, for 
example, of punches and presses, during a small 
part of the time required for the driving shaft or 
spindle of the machine to make a complete revolu¬ 
tion. If this work could be distributed over the 
entire period of the revolution, a comparatively nar¬ 
row belt would be sufficient to drive the machine; 
but a very broad and heavy belt would otherwise be 
necessary to overcome the resistance, if the belt 
only be depended on to do the work. It is, of 
course, in a sense, impossible to distribute the work 
of the machine over the entire period of revolution 
of the driving shaft of the machine, but by placing 
a large, heavy-rimmed wheel, a fly-wheel, on the 
shaft, the belt is given an opportunity to perform 
an almost uniform amount of work during the 
whole revolution. During the greater part of the 
revolution of the driving shaft the power of the 
belt is devoted to accelerating the speed of the fly¬ 
wheel. During that brief period of the revolution 
of the shaft when the work of the machine is being 
done, the energy thus stored up in the fly-wheel is 
given out at the expense of its velocity. The 

289 


290 SELF-TAUGHT MECHANICAL DRAWING 

energy a fly-wheel would give out if brought to a 
standstill would be (neglecting the weight of the 
arms and hub, as the efficiency of the wheel depends 
chiefly on the weight of the rim), expressed in 
foot-pounds, equal to the weight of the rim in 
pounds multiplied by the square of its velocity at 
its mean diameter in feet per second, and this 
product divided by 64.32, the-same as in the case 
of a falling body moving at the same velocity, as 
explained in the section on mechanics. 

Expressed as a formula this rule is: 

2g "64.32 

in which E = total energy of fly-wheel, 

W = weight of fly-wheel rim in pounds, 

V = velocity at mean radius of fly-wheel 
in feet per second, 

cj = acceleration due to gravity = 32.16. 

If the speed of the fly-wheel is only reduced, the 
energy which it would give out would be equal to 
the difference between the energy which it would 
give out if brought to a full stop, and that which 
it would still have stored up in it at its reduced 
velocity. Therefore, to find the energy in foot¬ 
pounds which a fiy-wheel will give out with an 
allowable loss of speed, subtract the square of the 
velocity of the rim in feet per second at its reduced 
speed from the square of its velocity in feet per 
second at full speed, multiply this difference by 
the weight in pounds, and divide the product by 
64.32. The result will give the loss of energy in 
foot-pounds. 


FLY-WHEELS 


291 


This long and cumbersome rule is expressed 
much more simply by the formula: 

") W 


E,= 


V 


64.32 


in which Ei = energy, in foot-pounds, fly-wheel 

gives out while speed is reduced 
from Vi to v ->, 

Vi = speed before any energy has been 
given out, in feet per second, 

V 2 = speed at end of period during which 
energy has been given out, in feet 
per second, 

W = weight of fly-wheel rim in pounds. 

This rule and formula may be transposed as fol¬ 
lows : To find the weight of a fiy-ivheel to give out 
a required amount of energy with an alloivable loss 
of speed, multiply the required amount of energy 
in foot-pounds by 64.32, and divide the product by 
the difference between the square of the velocity 
of the rim, at its mean diameter, in feet per second 
at full speed, and the square of its velocity in feet 
per second at its reduced speed; or, expressed as a 
formula, using the same notation as above: 

X 64.32 

V r - V f 

When the mean diameter of the fly-wheel is 
known, the velocity of the rim at its mean diameter 
in feet per second will be 

Di ameter in feet X 3.1416 X rev, per minute 

~ 60 


It is evident that in designing a fly-wheel for a 








292 SELF-TAUGHT MECHANICAL DRAWING 


machine, there is an opportunity for a wide range 
in the weight, from a wheel heavy enough, when 
once it has been brought to its full speed, to do, by 
means of the energy stored in it, the work without 
assistance from the belt, the belt being only just 
wide enough to restore the speed of the wheel in 
time for the next operation, to a wheel where the 
belt is wide enough to do the most of the work 
directly, the stored energy in the fly-wheel merely 
assisting it somewhat. Perhaps the best way would 
be to have the wheel heavy enough so that its 
stored energy could do the bulk of the work, the 
belt assisting it, and at the same time have the 
latter wide enough to quickly restore the speed of 
the wheel, so that, in case its velocity should be 
reduced beyond that calculated, there would be a 
margin of available power in the belt. 

Example .—Let it be required to design a fly¬ 
wheel for a press to cut off one-inch round bar 
steel, the press making 30 strokes per minute. 

Soft steel having a shearing resistance of about 
50,000 pounds per square inch, and a one-inch bar 
having an area of cross-section of 0.7854 square 
inch, the shearing resistance of the bar will be 
50,000 X 0.7854 = 39,270 pounds, or practically 
40,000 pounds. This resistance varies, however, 
during the process of shearing, being greatest near 
the beginning of the cut, and decreasing as the 
cutting progresses. In the case of a round bar it 
could not decrease uniformly, because of the shape 
of the cross-section. For the sake of getting the 
decrease in resistance as nearly uniform as possible, 
we will assume that the work of cutting off a one- 


FLY-WHEELS 


293 


inch round bar is the same as the work of cutting 
off a square bar of the same area; though this may 
not be quite exact, it would probably not be far 
out of the way. The length of the sides of a square 
of the same area as a given circle, is equal to the 
diameter of the circle multiplied by 0.886. There¬ 
fore, our equivalent square bar will be 0.886 of 
an inch square. The mean resistance to cutting, 
assuming that the resistance decreases uniformly 
as the cutting progresses, would be 40,000 -i- 2 ^ 
20,000 pounds. As the cutting operation continues 
through a space of 0.886 of an inch, the power 
required would be 20,000 X 0.886 = 17,720 inch- 
pounds, or 1476.6 foot-pounds. Let us plan to have 
the belt do one-fifth of the work of cutting direct¬ 
ly, leaving four-fifths to be done by the stored up 
energy of the fly-wheel. One-fifth of 1476.6 equals 
295.3. Subtracting this from 1476.6 leaves 1181.3 
foot-pounds to be supplied by the energy of the 
fly-wheel. As a preliminary calculation let us find 
what would have to be the weight of the wheel if 
it were to be placed upon the crank-shaft, the shaft 
which operates the plunger of the press. Assuming 
the mean diameter of the fly-wheel rim to be 4 
feet, the circumference would be 4 X 3.14 = 12.56 
feet, and, as the shaft makes 30 revolutions per 
minute, the velocity of the rim in feet per second 
would be: 


12.56 X 30 

60 


6.28 feet. 


If we expect the fly-wheel to suffer a loss of, 
say, 10 per cent, while doing its work, then its 
velocity at its reduced speed will be 6.28 — 0.628 = 



294 SELF-TAUGHT MECHANICAL DRAWING 


5.65 feet. The weight of the fly-wheel to give out 
1181.3 foot-pounds under these conditions will then 
be, according to the rule and formula already 
given: 

1181.3 X 64.3 2_ 75,981.2 ^75,981.2 _ .. ... 

6.28‘'-5.65- 39.44-31.92 7.52 

nearly. 


A wheel weighing 10,100 pounds would, of course, 
be out of the question; but as the energy increases 
as the square of the velocity, the weight may be 
very rapidly reduced by mounting the wheel upon 
a higher-speeded secondary shaft, connected with 
the crank-shaft by reducing gears. If the speed 
of the secondary shaft is to the speed of the crank¬ 
shaft as 6 to 1, the weight of the wheel, if the 
mean diameter be kept the same, will need to be 
only about one thirty-sixth of what it would need 
to be if mounted on the crank-shaft. At this^higher 
speed, however, it might be desirable to somewhat 
reduce the diameter of the wheel. Let us assume 
that the mean diameter be made 3 feet. If the 
ratio of speeds is 6 to 1, the wheel will make 180 
revolutions per minute, and the velocity of the rim 
in feet per second will be: 


3 X 3.14 X 180 
60 


28.3 feet. 


If the wheel suffers a loss of 10 per cent., its 
velocity at its reduced speed will be 28.3 - 2.83 = 
25.5 nearly. The weight of the wheel will then 
be: 


118L3 >^64.32 _ 75,981.2 
28.3 '- 25. 5 ' 150.64 


504 pounds. 







FLY-WIIB^ELS 


295 


As a cubic inch of cast iron weighs 0.26 pound, 
the wheel will contain 504 0.26 = 1938 cubic 

inches. The mean circumference of the rim in 
inches will be 3 X 12 X 3.14 = 113 inches. The 
cross-section of the rim will then be: 


1938 113 = 17.1 square inches. 


This would mean a rim about 4 by 4i inches. 
The outside diameter of the wheel would then be 
40 inches. 

We planned to have the belt do one-fifth of the 
work, and this we found to be 295.3 foot-pounds. 
If the crank has a radius of li inch, the cutter will 
have a stroke of 2i inches, and if the cutters over¬ 
lap each other one-quarter of an inch at the end of 
the stroke, the crank will have to swing through 
an angle of about 54 degrees in order to make the 
cutters advance the one inch necessary to cut off 
the one-inch bar, as a simple lay-out will show. 
The belt must then develop 295.3 foot-pounds while 
the crank swings through 54 degrees. It will then 
develop 295.3 ^ 54 = 5.5 foot-pounds, nearly, in one 
degree, and in a complete revolution it will develop 
5.5 X 360 = 1980 foot-pounds. As the press makes 
30 strokes per minute, the belt Vv^ill develop 30 X 
1980 = 59,400 foot-pounds per minute. If a driving 
pulley 18 inches in diameter is used, the belt speed 
in feet per minute will be: 


18 X 3 .14 X 180 
12 


848 feet. 


If a single thickness belt, one-inch wide, at 
1000 feet per minute, transmits 33,000 foot-pounds 




296 SELF-TAUGHT MECHANICAL DRAWING 


per minute, the same belt at 848 feet per minute 
will transmit tVfo as much, or 33,000 X 0.848 = 
27,984 foot-pounds. The width of belt necessary 
to transmit 59,400 foot-pounds per minute at this 
speed will then be 59,400 27,984 = 2.1 inches. 

No account has so far been taken of the power 
necessary to drive the machine itself. To allow 
for this the belt should evidently be not less than 
2J inches wide. A 3-inch belt would allow consid¬ 
erable of a margin of safety, and further calculation 
wall show that such a belt would develop, during 
about one-third of a revolution of the crank, the 
amount of energy which the fly-wheel had lost, so 
that, as the cutting operation takes about one-sixth 
of a revolution, the fly-wheel would be running at 
full speed for about one-half of a revolution of the 
crank, previous to the beginning of the cut, pro¬ 
vided that it had not suffered any greater reduction 
of velocity than the 10 per cent, planned for. 

If the press was employed doing punching the 
same method of procedure would be employed in 
the calculations, the area in shear in such a case 
being equal to the circumference of the hole mul¬ 
tiplied by the thickness of the plate. The end of a 
punch is usually made slightly conical or slightly 
beveling, the effect in either case being to increase 
the shearing action, and make the work of punch¬ 
ing easier. 


CHAPTER XIX 


TRAINS OF MECHANISM 

For obtaining high speeds without the use of 
unduly large driving pulleys or gears, for securing 
gain in power by sacrificing speed, for securing 
reversal of direction, or for obtaining some par¬ 
ticular velocity ratio between the driver and some 
part of the mechanism, pulleys, gears, worm-gears, 
or the like, may be substituted for direct acting 
driving-mechanisms. 

To Secure Increase of Speed.--Let a shaft making 
100 revolutions per minute be required to drive the 
spindle of a machine at 2000 revolutions per minute, 
the pulley on the spindle being 3 inches in diam¬ 
eter. If a direct drive were to be used, the pulley 
on the shaft would have to be as many times greater 
than the pulley on the spindle as 2000 is greater 
than 100, or 20 times. 

This would mean a pulley on the shaft 60 inches 
in diameter. Practical considerations, such as the 
weight of the pulley, size of hangers and the like, 
would make such a pulley out of the question. 

By interposing an intermediate countershaft be¬ 
tween the first shaft and the spindle of the machine, 
however, having pulleys of such size that the 
product of the ratio of the pulley on the first shaft 
and the one to which it is belted on the counter¬ 
shaft, multiplied by the ratio of the second pulley 

297 


298 SELF-TAUGHT MECHANICAL DRAWING 


on the countershaft and the pulley on the spindle 
to which it is belted is equal to the ratio which 
it is desired to have between the first shaft and 
the spindle, the same speed may be secured by the 
use of pulleys of convenient size. Thus, if the 
ratio between the pulley on the first shaft and the 
one on the countershaft is as 1 to 4, and the ratio 
between the driving pulley on the countershaft 
and the one on the spindle of the machine is as 
1 to 5, the product of these two ratios, 1 to 4 and 1 
to 5, is 1 to 20, and the arrangement will give the 



Fig. 204.—Reversal of Direction Obtained by Crossed Belt. 


required speed. The pulley on the spindle being 3 
inches in diameter, the driving pulley on the coun¬ 
tershaft will be 15 inches in diameter, and if the 
driven pulley on the countershaft is 4 inches in 
diameter the pulley on the first shaft to which it is 
belted will be 16 inches in diameter, instead of 60 
inches, as would be required with direct belting. 

If the spindle of the machine, instead of being 
driven were made the driver, as it would be if it 
were the armature shaft of a motor, then this ar¬ 
rangement would give gain in power with con¬ 
sequent loss of speed. 

To Secure Reversal of Direction.—In cases where 
shafts are belted together, reversal of direction of 


TRAINS OF MECHANISM 


•299 


rotation is secured by simply using a crossed belt 
instead of an open one, as shown in Fig. 204. 
When gears are used, reversal of direction of rota¬ 
tion follows as a natural condition of their meshing 
together, as shown in Fig. 205. In order that the 
two gears A and B shall rotate in the same direc¬ 
tion, it is necessary to separate them slightly, and 
interpose an intermediate gear, or idler, between 



Fig. 205.—Relative Direc- Fig. 206.—Influence of Idler 
tion of Rotation in a on Direction of Rotation. 

Pair of Gears. 


them as shown in Fig. 206. The rates of rotation 
of A and B with regard to each other is not affected 
by the idler gear, whether the idler be large or 
small. That this is so may be seen by direct exam¬ 
ination. If A is the driver, its circumference will 
impart to the circumference of C its own rate of 
motion, and C will in turn impart to B the same 
rate of motion, which is the same as it would have 
if in direct connection with A. 

If, now, another idler be interposed between A 
and B, making four gears in the train, A and B 
will again rotate in opposite directions. From this 
it will be seen that when a train is composed of an 


^00 SELF-TAUGHT MECHANICAL DRAWING 

even number of gears, the first and last members 
rotate in opposite directions; but when the train is 
composed- of an odd number of gears, the first and 
last members rotate in the same direction. 

In Fig. 207 is shown the mechanism used in 
engine lathes to secure either direct or reversed 
motion, by having the working train consist of 
either an even or an odd number of gears. In this 




Fig. 207.—Principle of Turn- Fig. 208.—Principle of Com- 

bler Gear. pound Idler. 

arrangement A is a gear on the head-stock spindle, 
and B is a. gear on a stud below. Pivoted on the 
axis of B is a triangular piece of metal, or bracket, 
shown in dotted lines, which can be swung back 
and forth by the handle E. Mounted on this 
bracket are the idler gears C and D, C being con¬ 
stantly in mesh with B, and D being in mesh with 
C. When it is required that B shall rotate in the 
same direction as A, the handle E is lowered until 
C meshes with A, The working train then consists 




TRAINS OF MECHANISM 


8Ui 

of three gears, A, C and B, D being out of mesh with 
A, revolving by itself, but not forming a part of 
the working train. When it is desired that B shall 
rotate in the opposite direction to A, the handle E 
is raised until D meshes with A, C being thrown 
out of mesh with it. The working train then con¬ 
sists of four gears, A, D, C and B, and the desired 
reversal is secured. 

The Compound Idler.—It has been shown that 
when a train consists of simple gears the relative 
rates of rotation of the first and last members re¬ 
main unchanged, regardless of the number or size 
of the idlers that may be interposed. When it is 
desired to secure a different rate of rotation be¬ 
tween two members of a train than that which 
they would have if meshing directly together, a 
compound idler is used, as shown in Fig. 208. Such 
a gear is used on many screw cutting lathes. For 
cutting threads up to a certain number per inch 
the screw cutting train consists of simple gears. 

A compound idler may then be introduced into 
the train, when without other change additional 
threads may be cut. If with screw cutting trains 
of simple gears a lathe will cut all whole numbers 
of threads up to 13 threads per inch, then, by adding 
a compound idler to the train, having its two steps 
in the ratio of 2 to 1, threads from 14 to 26 per inch 
(except odd numbers) may be cut with the same 
gears as previously used for cutting up to 13 threads 
per inch. If the compound idler forms an additional 
member of the train, the reversal of direction of 
rotation which would take place in the motion 
of the lead-screw of the lathe may be taken care of 


302 SELF-TAUGHT MECHANICAL DRAWING 


by the reversing gears between the spindle of the 
head-stock and the stud, previously described, and 
shown in Fig. 207. 

The Screw Cutting Train.—In Fig. 209 is shown 
the screw cutting mechanism found on engine 
lathes. The reversing mechanism shown in Fig. 




Fig. 210 . 


Figs. 209 and 210. —Arrangement of Lathe Change Gearing. 


207 is reproduced entire, and these gears—the gear 
A on the lathe spindle, the gear B on the stud, 
which is connected with A by the idlers C and D — 
are all permanent gears. These gears are usually 
on the inside of the head-stock as shown in Fig. 
210. The stud reaches through the head-stock, and 
on its outer end is the change gear F, connecting 
with the change gear G on the lead-screw of the 
lathe by means of the intermediate idler H, The 
idler II is mounted on a slotted swinging arm as 
shown, so as to allow of gears F and G being 





























TRAINS OF MECHANISM 


303 


replaced by others of such size as may be required 
to cut the particular screw desired. The carriage of 
the lathe, carrying the screw cutting tool, is driven 
directly by the lead-screw. On large lathes this 
screw is quite coarse, four threads per inch being 
common, while on smaller lathes a finer thread is 
used. The gear A on the spindle and the fixed gear 
B on the stud are sometimes of the same size, and 
sometimes of different sizes. 

The problem met with in screw cutting is to find 
what sizes change gears, F and G, must be used 
so that the lead-screw shall drive the carriage along 
one inch while the spindle of the lathe is making 
a number of revolutions equal to the number of 
threads to be cut per inch. Let us take as an 
example the assumed case of a lathe in which the 
lead-screw has 9 threads per inch, and in which 
the number of teeth in the gear on the spindle is 
to the number of teeth in the fixed gear on the stud 
as 3 to 4; required the size of change gears to cut 
23 threads per inch. Then, as the lead-screw has 
9 threads per inch, the spindle of the lathe must 
make 23 revolutions while the lead-screw is making 
9 revolutions. The method used in a previous 
chapter for obtaining the size of pulleys to give 
required speeds will give us the solution of this 
problem; if the speed of the first driving member 
of the train, together with the number of teeth or 
relative sizes of all other driving members be placed 
on one side of a vertical line, and the speed of the 
last driven member, together with the number of 
teeth or relative sizes of all other driven members 
be placed on the other side of the line, the product 


304 SELF-TAUGHT MECHANICAL DRAWING 


of the numbers on one side of the line multiplied 
together will equal the product of the numbers on 
the other side of the line multiplied together. The 
spindle of the lathe is, of course, the first driving 
member of the train, and the lead-screw is the 
last driven member. As the spindle is to make 23 
revolutions while the lead-screw makes 9 revolu¬ 
tions, 23 will be the first number on the side of the 
line on which the driving members are placed, and 
9 will be the last number on the side of the line on 
which the driven members are placed. Next, as 
the ratio between the sizes of the driving gear on 
the lathe spindle and the fixed gear on the stud 
below which it drives is as 3 to 4, these numbers 
will be placed against each other on opposite sides 
of the line. 

The ratio between the numbers of teeth or sizes 
of the two change gears, F and G, whose sizes it 
is required to find, being unknown, may be said to 
be as 1 to the unknown number x. These numbers, 
1 and X, are now placed on their proper sides of 
the line, and the problem appears as shown below. 
The size of the idler gear H does not enter into the 
question, because, as has been previously shown, a 
simple idler gear does not affect the relative rates 
of rotation of the gears between which it transmits 
motion. 


Speed of spindle • 23 

Ratio of size of spindle gear 3 
Ratio of number of teeth 
in change gear F 1 


4 to size of fixed stud gear. 

X to number of teeth in 
change gear G 
9 speed of lead-screw. 


69 = 36a; 





TRAINS OF MECHANISM 


305 


Multiplying together the numbers on both sides 
of the line gives the equation 69 = 36a;. It is evi¬ 
dent that if 69 equals 36a;, x must be equal to 69 
divided by 36, or f|. The ratio between sizes of 
the gear F and the gear G is then as 1 to f |. 
Eliminating the fraction by multiplying both terms 
of the ratio by 36 gives the ratio as 36 to 69. If, 
then, F has 36 teeth, and G has 69 teeth, the lathe 
will cut the required number of 23 threads per 
inch. 

In Fig. 211 is shown how a compound idler gear 
is sometimes used in a screw cutting train. The 



Fig. 211.—Compound Gearing. 


change gear G and the idler H have long hubs on 
one side. When it is desired to cut finer threads 
than what the gears E and G with the idler H will 
give, H and G, are put on with the long hubs 
toward the lathe, throwing them out of line with 
E. The gear E then meshes into the large step of 
/, the small step of I meshes into H, and H meshes 


30G SELF-TAUGHT MECHANICAL DRAWING 


into G. The ratio between the large and the small 
steps of 1 must then be taken into account in the 
calculation. For cutting the coarser threads H and 
G are put on with the short hubs toward the lathe, 
bringing them into line with E. The idler I is also 
turned over, so that its large step is on the outside 
and out of line with E and H. It is then swung 
back out of the way. 

When the gearing is fully compounded the two 
gears at I are separate from each other but keyed 
together on the same stud and mounted in the 
same manner as shown in Fig. 211. By varying 
the sizes of these gears, almost any screw thread 
may be cut within reasonable limits. In this case, 
of course, there are four gears to be determined in 
our calculations. Simplified rules are given in the 
following for this case, as well as for the regular 
simple trains. 

Large lathes are provided with change gears for 
cutting threads from about 2 to about 20 threads 
per inch, smaller lathes being provided with gears 
for cutting from about 3 or 4 to 40 or 50 threads per 
inch, in either case including a pair of gears for 
cutting llj threads per inch, this being the stand¬ 
ard thread for iron pipes from one to two-inch sizes 
inclusive. The smaller lathes would also naturally 
be provided with gears for cutting 27 threads per 
inch, this being the number of threads on J-inch 
iron pipes. 

Simplified Rules for Calculating Lathe Change 
Gears.—The following rules for calculating change 
gears for the lathe have been published by Ma¬ 
chinery (Reference Series Book No. 35, Tables 


TRAINS OF MECHANISM 


307 


and Formulas for Shop and Draftingroom), and 
are here given because of their concise form and 
simplicity. 

Rule 1. —To find the ‘‘screw-cutting constant” of 
a lathe, place equal gears on spindle stud and lead- 
screw ; then cut a thread on a piece of work in the 
lathe. The number of threads cut with equal 
gears is called the “ screw-cutting constant” of 
that particular lathe. 

Rule 2. —To find the change gears used in simple 
gearing, when the screw-cutting constant as found 
by Rule 1, ahd the number of threads per inch to 
be cut are given, place the screw-cutting constant 
of the lathe as numerator and the number of threads 
per inch to be cut as denominator in a fraction, and 
multiply numerator and denominator by the same 
number until a new fraction is obtained represent¬ 
ing suitable numbers of teeth for the change gears. 
In the new fraction, the numerator represents the 
number of teeth in the gear on the spindle stud, 
and the denominator, the number of teeth in the 
gear on the lead-screw. 

Rule 3. —To find the change gears used in com¬ 
pound gearing, place the screw-cutting constant as 
found from Rule 1 as numerator, and the number 
of threads per inch to be cut as denominator in a 
fraction; divide up both numerator and denomi¬ 
nator in two factors each, and multiply each pair 
of factors (one factor in the numerator and one in 
the denominator making a pair) by the same num¬ 
ber, until new fractions are obtained, representing 
suitable numbers of teeth for the change gears. 
The gears represented by the numbers in the new 


308 SELF-TAUGHT MECHANICAL DRAWING 

numerators are driving gears, and those in the 
denominators driven gears. 

Two examples, showing the application of these 
rules, will be given in the following. 

Example i.—Assume that 20 threads per inch are 
to be cut in a lathe having a “screw-cutting con¬ 
stant,'’ as found by the method explained in Rule 
1, equal to 8. The numbers of teeth in the avail¬ 
able change gears for this lathe are 28, 32, 36, 40, 
44, etc., increasing by 4 up to 96. 

By applying Rule 2, we have then: 

A _ 8 X_4 _ 32_ 

20 20 X 4 80 

By multiplying both numerator and denominator 
by 4 we obtain two available gears having 32 and 
80 teeth. The 32-tooth gear goes on the spindle 
stud and the 80-tooth gear on the lead-screw. It 
will be seen that if we had multiplied by 3 or by 5 
instead of by 4, we would not have obtained avail¬ 
able gears in both numerator and denominator, as 
8X3 would have given 24 and 20 X 5 would have 
given 100, both of which gears are not in our given 
set of gears. The proper number by which to 
multiply can be found by trial only. 

Example 2 .—Assume that 27 threads per inch are 
to be cut on the same lathe as assumed in Example 1. 

In this case the calculation must be made for 
compound gearing, as so fine a pitch could not be 
cut by simple gearing in this lathe. By applying 
Rule 3 we have: 

_8 _ 2X4 _ (2 X 20) X (4 X 8) _ 40 X 32 

27 3 X 9 (3 X 20) X (9 X 8) 60 X 72 





TRAINS OF MECHANISM 


309 


The four numbers in the last fraction give the 
numbers of teeth in the required gears. The gears 
in the numerator (40 and 32) are the driving gears, 
and those in the denominator (60 and 72) are the 
driven gears. 

It makes no difference which one of the driving 
gears is placed on the spindle stud or which one of 
the driven gears is placed on the lead-screw. 

Back-Gears.—Nearly all engine lathes and many 
other machine tools are provided with a set of re- 



Fig. 212.—Principle of Back-Gearing. 


ducing gears, called back-gears, by means of which 
double the range of speeds that can be obtained by 
direct driving may be given to the spindle of the 
machine. Fig. 212 illustrates such a set of gears, 
and the method of applying them to the machine. 
The large gear A is fastened to the spindle of the 
machine, but the cone pulley, with the gear B 
attached to it, is loose on the spindle. The back- 











































































































310 SELF-TAUGHT MECHANICAL DRAWING 


gear shaft with gears C and D is mounted in 
brackets on the back side of the head-stock, and 
is provided with eccentric bearings, by means of 
which the gears on it can be thrown into or out of 
mesh with the gears on the head-stock spindle. 
When direct driving is desired, the back-gears are 
thrown back, out of the way, and the cone pulley 
and the large gear are clamped together by means 
of a screw pin or stud passing through the gear 
into the cone. They then revolve together as one 
piece. 

Let us assume the case of a lathe having a cone 
with four steps, the largest step being 6 inches in 
diameter, and the smallest 4 inches in diameter, 
with the intermediate steps in proper proportion. 
If the cone pulley on the countershaft is of the 
same size as the one on the spindle, then, if the 
countershaft runs 300 revolutions per minute, direct 
driving will give about the following speeds to the 
spindle: 450, 345, 260 and 200. Let it now be 
required to find the sizes of gears to be used so 
that with the back-gear driving, a proportionately 
slower rate of speeds may be obtained. We may 
solve the problem by giving to the gears some 
arbitrary sizes, and finding what speeds such sizes 
will give, and then modify these sizes until the 
required speeds are obtained. For trial purposes 
let us make the pitch diameter of the gear A the 
same as the diameter of the large step of the cone 
pulley, or 6 inches, and the pitch diameter of the 
gear B the same as the diameter of the small step 
of the cone pulley, or 4 inches. Arranging driving 
and driven members on opposite sides of a vertical 


TRAINS OF MECHANISM 


311 


line, the speed of the first driving member of the 
train, the countershaft, being 300, the required 
speed of the last member, the lathe spindle, being 
represented by x, and having the belt on the largest 
step of the countershaft cone so as to obtain the 
highest speed with back-gears, gives an arrange¬ 
ment of the case as below. The sizes of the back- 
gears are the same as those on the lathe spindle, 
the gear C being 6 inches in pitch diameter, and 
the gear D 4 inches in pitch diameter. 

Speed of countershaft 
Pulley on countershaft 
Gear B on lathe 
Back-gear D 


X = 


300 

6 

4 

4 


4 Pulley on lathe 
6 Back-gear C 
6 Gear A on lathe 
X Speed of spindle 


28,800= 144a; 
2 8 ,800 
144 


= 200 . 


From this it is seen that with the sizes of the 
gears as above, the highest speed with back-gears 
would be the same as the lowest speed without 
the back-gears. This, of course, would be useless 
duplication of speeds. 

For another trial we will make the sizes of the 
gears B and D each 3i inches in pitch diameter. 
The calculation then becomes: 


Speed of countershaft 
Pulley on countershaft 
Gear B on lathe 
Back-gear D 


300 

6 

3.5 

3.5 


4 Pulley on lathe 
6 Back-gear C 
6 Gear A on lathe 
X Speed of spindle 


22,050 = 144a; 
22,050 


144 


= 153, nearly. 


X 






312 SELF-TAUGHT MECHANICAL DRAWING 

A speed of 153 revolutions per minute for the 
fastest back-gear speed follows quite regularly the 
series of speeds which the direct drive gives. 

Instead of using the pitch diameters of the gears 
in making the calculations the number of teeth 
which the gears would have, the pitch being first 
decided on, might be used. In this manner it is 
possible to make slight changes in the diameters of 
the gears without bringing troublesome fractions 
into the calculations. 

Many lathes and other machine tools have trains 
of mechanism much more complicated than any 
here shown, but the method of procedure here 
outlined can be applied to all of them. 


CHAPTER XX 


QUICK RETURN MOTIONS 

In a large class of machinery the work is done 
during the forward motion of a reciprocating part; 
the return of the part to its starting point is then 
a question of time. The quicker the part can be 
returned to its starting point, the more efficient 
becomes the machine. When the stroke is long, as 
in the case of the bed of an iron planer for large 
work, this rapid return motion is usually obtained 
by means of shifting the driving belt onto a return 
pulley so arranged that a higher ratio of speed is 
procured; but in other cases, where the recipro¬ 
cating motion is shorter, and the stroke is actuated 
by means of a crank, the actuating mechanism is 
made such that the crank gives a slow forward 
and a quick return motion to the reciprocating 
part. Iron planers for small work, shapers, and 
the like, and some classes of engines and pumps, 
use such quick return motions. Below are described 
the principal devices used for such purposes. 

Fig. 213 shows a method of securing a quick 
return by having the axis of the crank outside of 
the path of the reciprocating end of the connecting- 
rod. Let A be a crank, the crank-pin of which, a, 
acting upon the connecting-rod B represented by 
the heavy line, causes the block h to move back and 

313 




314 SELF-TAUGHT MECHANICAL DRAWING 


forth in the path CD. When the crank is in the 
position shown the block is at the extreme' left of 
its stroke, the connecting-rod and crank being in 
the same straight line, the center line of the con¬ 
necting-rod coinciding with the axis of the crank. 
As the crank swings downward, the block b is 
driven to the right; but an examination of the 
illustration will show that the crank must make 



more than a half revolution before it again forms 
a straight line with the connecting-rod, which it 
will do when the block has reached its extreme 
position to the right. As, therefore, the block 
makes its movement to the right while the crank 
is swinging through the lower angle included be¬ 
tween these two positions, and as it makes its 
return stroke while the crank is swinging through 
the upper angle included between these same two 
positions, the time of the forward stroke of the 
block will be to the time of its return stroke as 
the lower angle is to the upper angle. 







QUICK RETURN MOTIONS 


315 


The upper angle being the smaller of the two, 
the block has a quick return motion. To secure 
ease of motion to the block as it starts on its stroke 
to the right, the angle ahC, the angle which the 
connecting-rod makes with the path of the block, 
should not be more than about 45 degrees. 

To design a quick return motion of this type, lay 
out a horizontal line ah. Fig. 214, and on it mark 
off cb equal to the required length of stroke. From 
c draw the line cd of indefinite length at such an 


d 



obliquity that the angle acd shall not be more than 
45 degrees. From h draw the line he at the angle 
required to give the desired quick return. The 
intersection of these two lines at / will be the axis 
of the crank. The length hf will be seen by re¬ 
ferring back to Fig. 213 to be equal to the length 
of the crank plus the length of the connecting-rod. 
The length of cf will be seen to be equal to the 
length of the connecting-rod minus the length of 
the crank. If in a given case the length ch is 
made 12 inches, and cf is found to be 10 and hf 21 
inches, which they would be if the angles were as 



316 SELF-TAUGHT MECHANICAL DRAWING 

shown in Fig. 214, then, letting x represent the 
length of the connecting-rod and y the length of 
the crank, we would have x + y = 21 inches, and 
X — y = 1^ inches. Adding the left-hand and the 
right-hand members, respectively, of these two 
equations, we would have x-\-y-\-x-y = 21-¥l0 
= 31 inches. As 1 y - y= 0 we may eliminate 
these expressions, and the equation will read 2x ’= 
31 inches, and x, the length of the connecting-rod, 
will thus be 15i inches. The length of the crank 
will then be 21 inches (the length of hf) minus 15i 
inches, or 5i inches. 

It will be seen that if the length of the stroke is 
made variable by having the crank-pin, a, adjust¬ 
able to different positions on the crank A, Fig. 213, 
the difference between the time of the forward 
and of the return stroke of the sliding block 6 will 
be lessened, because the two positions which it 
will occupy at the extremes of its stroke will be 
nearer together, and the lower and upper angles 
which the crank passes through in giving to the 
block its forward and return movements will be 
more nearly equal. 

Fig. 215 shows a quick return motion device 
especially adapted to cases where the horizontal 
space is limited, and which is much used on shapers. 
The illustration shows a shaper in outline. The 
ram of the shaper is given its forward and return 
motion by means of the rocking arm A, which 
swings on a fulcrum at B. The rocking arm is 
given its motion by means of a crank-pin on the 
disk C, the pin engaging in a sliding block which 
travels in a slot in the arm A. 


QUICK RETURN MOTIONS 317 

Let BC and BD, Fig. 216, represent the extreme 
positions of the rocker arm A. Draw the lines OF 
and OG from the center of the crank disk at 0 at 
right angles to BC and BD, It is evident that in 
order that the crank, on its upper sweep, shall 



Fig. 215.—Diagram of Quick Return Arrangement 

in a Shaper. 


move the rocker arm from C to D, it must move 
through the arc FAG, while to return the»arm 
from D to C, on its lower sweep, it must move only 
through the lower arc FG. The time of the return 
motion will therefore be to the time of the forward 
motion as the lower arc or angle FG is to the arc 



























318 SELF-TAUGHT MECHANICAL DRAWING 

or angle FAG. If the crank is shortened so as to 
give a shorter stroke to the ram of the shaper, 
then the rocker arm will swing through a smaller 
angle, as from H to I, and lines drawn from 0 at 


A 



Fig. 216.—Diagram of Speed Ratios in Shaper Motion. 

right angles to HB and IB will be more nearly in a 
straight line than OF and OG. There will, there¬ 
fore, be less difference between the time of forward 
and return motions on short strokes than on long 
ones. 









QUICK RETURN MOTIONS 


319 


The Whitworth Quick Return Device.—Let A, 
Fig. 217, be a slotted arm revolving on its axis at 
B. Above A is the driving crank C, having a pin 
engaging in the slot at the left in the arm A. The 
slot at the right in the arm A is provided for an 
adjustable stud which drives the reciprocating 
parts, through the medium of the connecting-rod 



D. It will be seen that, as shown, the connecting- 
rod is at the extreme right of its motion, forming 
as it does a straight line with the revolving arm 
A, which latter is at the same time at right angles 
with the center line cd. It will be seen that in 
order that the arm A may move through half a 
revolution so as to bring the connecting-rod to the 
extreme left of its motion, it will be necessary for 
the actuating crank C to revolve either through the 


















320 SELF-TAUGHT MECHANICAL DRAWING 


upper angle x or through the lower angle y, so as 
to form again the same angle with the center line 
cd, but at the right of it, as it is now shown form¬ 
ing with it at the left. The forward and return 
motions will, therefore, be to each other as the 
angle x is to the angle y. To design a quick return 
motion of this type it is, therefore, necessary to 
first lay out the angles x and y of such relative 
sizes that cc is as many times greater than y as the 
time of the forward motion is to be greater than 
the time of the return motion, having them, of 
course, central on the line cd. The distance apart 
of the fulcrums of the crank C and of the revolving 
arm A will be partly determined by the sizes of 
their shafts. The location of the crank-pin, de¬ 
termining the length of the crank, will then be at 
the intersection of the horizontal center line of the 
revolving arm A with the dividing line e/between 
the angles x and y. The length of the crank must, 
of course, be sufficient so that the crank pin will 
swing under the hub of the arm A, and the length 
of the crank-pin slot in A must be sufficient for 
the motion of the pin relative to the arm. 

It will be noticed that, unlike the two preceding 
quick return devices, varying the stroke of the 
reciprocating parts does not alter the relative time 
of the forward and return motions; for such change 
does not affect the angles x and y upon which the 
time of the forward and return motions depends. 
If, however, the length of the crank C is varied, 
then the, angles x and y are altered, and the time 
of the forward and return motions will be affected. 

It will be seen upon examination that with the 


QUICK RETURN MOTIONS 


321 


construction shown the revolving arm A must be 
made in two parts, one at each end of its shaft, in 
order to avoid interference of the parts of the 
mechanism with one another as they revolve. This 
trouble is overcome by replacing the crank C with 
a crank disk which fits over and revolves upon a 
fixed stud or hub large enough to receive the stud 
at B upon which the arm A revolves. 

The Elliptic Gear Quick Return.—If two ellipses 
of equal size. Fig. 218, having foci at w and x and 


9 



Fig. 218.—Quick Return Motion by Means of Elliptic Gears. 


at y and 2 :, be placed in contact with each other 
with their long diameters forming a continuous 
straight line as shown; then if the ellipses are 
caused to revolve freely upon their correspond¬ 
ing foci, w and y, they will roll upon each other 
perfectly, without slipping. From the nature 
of an ellipse as shown by its construction with a 
thread and pencil (see Chapter III, Problem 13) it 
will be seen that if the ellipse at the left were 
being formed in this manner and the pencil were 
at D, the intersection of the circumference of the 






















322 SELF-TAUGHT MECHANICAL DRAWING 


ellipse with the long diameter, the length of the 
thread would be equal to the sum of the distances 
wD and Dx, But the distance Dx is the same as 
the distance Di/; therefore, the length of the thread 
would be equal to the distance wy, the distance 
between the foci upon which the ellipses are re¬ 
volving. If, now, the ellipses are revolved until 
the points A and B, vertically over the foci x and 
y, are in contact with each other, the sum of the 
distances wA and By will be equal to the distance 
between the foci w and y, for their sum is equal to 
the length of the thread, and the length of the 
thread is equal to wA plus Ax, and Ax is equal to 
By, as points A and B are both vertically over the 
foci of the ellipses. In a similar manner any pair 
of points may be selected on the two ellipses equally 
distant from the point D. The distance from the 
point on the ellipse at the left, to the focus w, will 
be equal to the length of the thread at the left of 
the pencil, and the distance from the point on the 
ellipse at the right, to the focus y, will be equal to 
the length of the thread at the right of the pencil, 
and their sum will be equal to the distance between 
the foci w and y. This distance between the foci 
w and y will be seen on further examination to be 
equal to the long axis of the ellipse. This property 
of the ellipse has been taken advantage of to secure 
a quick return motion to a reciprocating part of a 
machine. If in Fig. 218 the two ellipses represent 
the pitch lines of elliptic gears; with the gear at 
the left as the driver with a uniform motion, the 
one at the right will have an ununiform motion. 
If, now, a crank is mounted on the same shaft as 


QUICK RETURN MOTIONS 


323 


the driven elliptic gear, the crank having its center 
line at right angles to the long axis of the ellipse, 
and this crank actuates a sliding block back and 
forth in the direction of the center line of the two 
gears, then this block will have a slow motion in 
one direction, and a quick motion in the other 
direction. If, now, the gears are revolved from the 
position in which they are shown until A and B 
are in contact, the gear at the right will have made 
a quarter of a revolution and the sliding block will 
be at the extreme right of its stroke; but while 
this gear has made a quarter of a revolution, the 
driving gear has revolved through the angle AwD 
only. If, now, the gear at the right is revolved 
another quarter of a turn, the points E and F will 
be in contact, and the crank will be directed ver¬ 
tically upward. The driving gear will, however, 
have revolved through the angle AwF. The forward 
and return motions of the sliding block will, there¬ 
fore, be to each other as the angle AwF is to tne 
angle AwD. In designing a pair of elliptic gears, 
therefore, the first thing to do is to determine the 
size of the angle Awx. To find the distance be¬ 
tween the foci tv and x first lay out on a large scale 
a triangle similar to the triangle Awx. Then the 
sum of its hypothenuse and the perpendicular will 
be to the length of its base as the sum of wA and 
Ax (the long axis of the ellipse) is to wx, the dis¬ 
tance between the foci of the ellipse. The length 
of the short axis may then be found by reversing 
Problem 13, Chapter III. The problem may be 
solved even more accurately by the rules given for 
the solution of right-angled triangles. The length 


324 SELF-TAUGHT MECHANICAL DRAWING 


of wA will be to Ax as 1 is to the sine of the angle 
Awx. Dividing the long axis of the ellipse into 
two parts in this proportion gives the length oi wA 
and Ax. The length of wx will then be equal to 
the length of Aw multiplied by the cosine of the 
angle Awx. Then to find the short axis of the 
ellipse, divide the distance wx into two equal parts 
and construct the triangle wgh. The length wh 
will be half of the distance between the foci, and 
the length of wg wiW be half of the long axis. The 
length gh, half of the short axis, may then be found. 

Calculations made in this manner give the follow¬ 
ing proportions to ellipses for quick return ratios 
as indicated in the first column: 


Ratio of Forward 
to Return Motion. 

Long Axis. 

Short Axis. 

Distance Between 
Foci. 

2 to 1 

1.000 

0.963 

0.268 

2h to 1 

1.000 

0.936 

0.351 

3 to 1 

1.000 

0.910 

0.414 

4 to 1 

1.000 

0.860 

0.509 

5 to 1 

1.000 

0.817 

0.577 


There appear to be two difficulties with elliptic 
gearing. The first is that if a high quick return 
ratio is attempted, so as to make considerable dif¬ 
ference between the long and the short axes, the 
obliquity of the action of the teeth upon each 
other, and the consequent great amount of friction 
between the teeth as they come together, becomes 
so great as to be troublesome. This may, to a con¬ 
siderable extent at least, be overcome by using a 
train of gears, each gear but slightly elliptic, in 
place of one pair of decidedly elliptic form. Thus 













QUICK RETURN MOTIONS 


325 


a train of three gears having their long and short 
axes in the proportion required to give a quick 
return of 3 to 1, with one pair of gears, will give 
a quick return of 9 to 1. If three gears of the 4 to 
1 proportion are used, a quick return of 16 to 1 
will result. 

The second difficulty is that of correctly cutting 
the teeth. To work properly, the teeth should be 
cut on a machine having a special elliptic gear 
cutting attachment, otherwise the gears are likely 
to be expensive and unsatisfactory. Such an ellip¬ 
tical gear cutting arrangement is described, and 
the subject of elliptic gearing is quite fully dis¬ 
cussed, in Grant^s treatise on gearing. Not being 
within the territory of this elementary treatise on 
machine design, the subject cannot here be dealt 
with in detail. 


CHAPTER XXI 

THE TECHNIQUE OF MECHANICAL DRAFTING 

By C. W, Reinhardt 

The preceding chapters form as it were a well 
rounded outline of a treatise on the subject of 
Mechanical Drawing and Machine Design. The 
different branches as designated by chapter head¬ 
ings are compactly, yet quite comprehensively 
handled. To the writer it appears, however, as if 
the illustrations were not exactly done justice to, 
as if in some way the text matter were indeed well 
handled, but that the accompanying illustrations 
were more or less indifferently treated and some¬ 
what slighted. In saying this, the writer would 
not at all try to imply that the drawings in ques¬ 
tion were not correctly drawn. He would only 
. state that these illustrations could in every case 
have been presented the same size, but a different 
treatment would enhance their legibility and 
greatly improve their appearance. Instead of 
engraved lines and printed letters and numerals, 
each cut should consist of a reduced facsimile of 
an actual drawing; lines drawn in their relative 
thickness, the lettering also done by hand, as it 
should be on a real drawing. To this statement 
the objection might be raised, that the ordinary 
full-size drawing does not lend itself readily for the 
purpose of book illustration. While this might be 

326 


TECHNIQUE OF MECHANICAL DRAFTING 327 

partly true it would be only for the reason that 
such a drawing is not executed according to 
recognized and well-tested rules, by the observance 
of which a good reproduction from the original, 
reduced in size though,- could be obtained. 

Furthermore, it is also obvious that in every¬ 
day business pursuits the carefully worded docu¬ 
ment, well written either in ‘^Long Hand,” or by 
typewriter, receives more consideration than the 
other which is more or less indifferently handled, 
although both may possess equal merit. Just so, 
the well executed and lettered drawing will 
compel attention above its carelessly done com¬ 
petitor, yet their merits may be equal. 

In the following the writer has endeavored to 
present the correct and most advantageous - 
method of constructing neat and legible drawings, 
which are comprehensive even to the layman. 

The matter can, however, be only handled in 
outline, as it were, as lack of space will not permit 
any more detailed treatment. Any one wishing 
to study this subject more extensively, is referred 
to the author’s: Lettering for Students, Engineers 
and Draftsmen,” The D. Van Nostrand Company, 
Publishers, New York, as also to Reinhardt’s 
Kinks and Wrinkles,” published by The Norman 
W. Henley Publishing Co., New York. 

The matter of outline shading for drawings has 
received passing notice in a preceding chapter. 
Where such is employed, a decidedly heavy shade¬ 
line should be used, giving a good contrast with 
the lighter outlines and invisible,” as also 
auxiliary and dimension lines. See Fig. 219, as 


328 SELF-TAUGHT MECHANICAL DRAWING 

contrasted with Fig. 220, where the shade-lines are 
omitted; yet the visible outlines receive quite a 
heavy rendering, so as to show sufficient difference 
between these and the other class of lines. 


b= ^di-^,h=bh'=d 



H- 


H \* h’A 



/ 



• 


A i 1 

• “<5 i 

Y 1 1 

fir 


f 

1 

Li- 

k- 

... j 


i. 


-I . 


Fig. 220 

A warning should be given here. Ordinary 
light outlines should never be drawn too light; 
always remember, that no matter how thin the 
line, a solid ridge of black ink should be on top of 
it. Never screw up the ruling pen so tight, that 
the ink will be forced out underneath the nibs, 
thus causing gray lines, which will not properly 
blueprint or photograph. Avoid also lines shown 
under Fig. 221. The broken lines should always 


Fig. 221 

consist of regularly sized dashes or dash-and-dots 
with regular spaces between. In that respect 



















































TECHNIQUE OF MECHANICAL DRAFTING 329 

the careless draftsman is sinning the most. Note 
Fig. 222. As already stated, outline shading can 
be used on assembled drawings with advantage as 
it assists the eye in reading such a sheet more 


Incorrect 


Correct 



Incorrecf 




Correct 
Fig. 222 


rapidly. Shade-lines, as a rule, should be con¬ 
sidered a substitute for actual shadows cast. If 
this maxim is borne in mind, it will be easy to 
determine the location of such lines. The shade¬ 
lines will, therefore, be always located on the out¬ 
side of an object, which ordinarily would cast a 
shadow. See Fig. 223. Shading of circles and 


O 


Shade Lines 
Correch' 

Fig. 223 

circular arcs may be effected by shifting the centers 
of the compass or bow pen a trifle to the ‘‘south¬ 
east’^ as indicated by minute circles in Fig. 224, 
A and B. Tapering shade-lines are shown some¬ 
what exaggerated in Fig. 225, Y, Y, which is 
really self-explanatory. 

Objects, which for some reason are shown 



Shade Lines 
Incorrecl' 




























330 SELF-TAUGHT MECHANICAL DRAWING 


broken or cut open, so as to reveal interior parts 
and arrangements, should be consistently section- 
lined. Section-line ruling should show somewhat 
thinner than ordinary outlines, and should further¬ 
more be not drawn too close. Some very con¬ 
venient symbols and standards for section-lining 



are contained on Plates I to V of Reinhardt^s 
“Kinks and Wrinkles.” There a complete set 
of draftsman’s standards for that class of work is 
shown, as well as directions for freehand section¬ 
ing, where such is needed, as for instance, to in¬ 
dicate wood, earth, rock, gravel, etc. 

Where thin parts of metal are indicated in 
section, the same may be shown in solid black, as 
a matter of convenience. This is the rule in 
structural drafting. Larger surfaces in conjunc¬ 
tion with these black sections are nevertheless 
section-lined in the regular way. 

Curved surface or cylinder shading may some¬ 
times be used to advantage, either for a specially 
finished drawing, or else to denote rounded sur¬ 
faces on a drawing, which otherwise could not well 


















TECHNIQUE OF MECHANICAL DRAFTING 331 

be recognized as such. For special directions or 
rules governing this style of shading, we would 
again refer the reader to above-mentioned pub¬ 
lication. Note differences in appearance of same 
casLng in Fig. 226. Where parts of a drawing 



are not desired to be depicted at full length, they 
are shown broken off, the “break” line suggesting 
the material described, as for instance a fairly 
smooth wavy line might indicate a break in 
“concrete,” whereas a jagged or serrated break 
will show “wood,” see Fig. 227. An excellent 

T S/c/e l^/e^v 
P/an 

Round Bar. Pipe or Tube. R.R;Rail. 





L 


Angle. 

zm 


Concrele. Wood. 


I-Beam. Z-Bar. Channel. 

Fig. 227 


way of imparting additional information concern¬ 
ing the shapes of objects shown broken off, is to 
suggest their shape (almost pictorially) in the 
break, as shown in a few cases under Fig. 227. 































































































CHAPTER XXII 

FREEHAND LETTERING FOR WORKING DRAWINGS 

The subject of lettering for Working Drawing 
has during the last twenty years received an 
unusual amount of attention. The consensus of 
opinion, as advanced by a score of text-books upon 
the subject, is that the so-called one-stroke style 
of lettering, originally devised and published in 
Reinhardt’s “Lettering,” is the most practical 
and easiest constructed. In the present, there¬ 
fore, we shall consider this one style of script. 
For a complete treatise upon the subject, we would, 
of course, refer the reader to above-mentioned 
book, as lack of space forbids the presentation of 
anything more than fragmentary in character. It 
is, of course, understood that good lettering on a 
poorly executed drawing may be a redeeming 
feature, and will make the sheet presentable, 
whereas on the other hand, a poorly lettered draw¬ 
ing, no matter how well executed otherwise, will 
surely be marred in appearance. So in order to 
facilitate the proper practice in forming letters, the 
author devised a system of arrows and numerals, 
denoting direction as well as sequence of stroke, 
on each letter or numeral. By following these 
hints faithfully a good “hand” in lettering will 
very soon be developed by the student. 

We naturally would use pencil guide-lines for 

332 


LETTERING FOR WORKING DRAWINGS 333 

lettering; for working on transparent material, 
such as tracing paper or cloth, we employ ruled 
cross-section paper, which would give the sizes of 
letters as well as the vertical spacing. The 
governing proportions for this style of lettering are 
3 to 5, i.e., the smaller lower-case letters, such as 
r or n, should be 3 units high, as against the height 
of the higher lower-case, such as 1 or h, as well as 
the capital letters, which are supposed to be 5 units. 
Ordinary lettering on a working drawing should, 
as a matter of fact, never be made less in height 
than tV inch for smaller lower-case letters; wher¬ 
ever possible, however, this size should be doubled, 
so as not to make lettering appear too cramped. 
The slant of the letters should be 1 horizontal to 
23^2 vertical. 

We have arranged the following letters in 
groups so as to give a progressive arrangement as 
regards case in construction. The direction and 
sequence of strokes is indicated by arrows and 
numerals. The downstroke of letters should 
never be exactly straight, but a very slight reverse 



Fig. 228 



(/i c e s 8 0 0l 



Fig. 229 


Fig. 230 


334 SELF-TAUGHT MECHANICAL DRAWING 

curve such as shown in Fig. 229. In the next 
group, Fig. 230, the general slant of the ellipses of 
the o, c, e, and of the a and d should be contrasted. 
The axes of the latter two are at an angle of 45° 
with the horizontal whereas the axes of the 
former are parallel with the downstroke, or 2}/2 
to 1. Care should be taken to have the bases of 
the a and d spread sufficiently, so as to give ap¬ 
parent stability. In group Fig. 231, the numerals 


q g p b 2 3 5 69 y w 



231 


shown deserve special attention, as they are of 
somewhat unusual shape and they should be very 
carefully studied and practiced faithfully, until 
students^ work compares well with original shown 
here. The capital letters shown under Figs. 232, 
233, 234, are quite easy to imitate, and present no 


I LT E F H K N M A 



Fig. 232 


V WXZ.U YJ D B R 



Fig. 233 


LETTERING FOR WORKING DRAWINGS 335 


0 Q G S 

Fig. 234 


special difficulty. As a precaution, the writer 
would advise, to get the corners of tops and bot¬ 
toms of L, E, F and a few others very pointed, as 
shown in Fig. 235, ‘'A,’’ so as not to get the effect 


LEF LEF 

-A- -B- 

Fig. 235 


shown under same figure. 

The flexibility of this system of lettering is 
demonstrated in Fig. 236 also, where letters of 


Elngineering 

Cylindrical 
Fig. 236 


same height may be expanded or contracted, as 
the necessity arises. 

The slanting lettering described thus far is best 
employed for all descriptive and dimensioning 
matter, notes, etc., on drawings, for captions 
which indicate the different portions of a drawing, 
upright cap and lower-case lettering is employed. 
Main titles, as a rule, are drawn at a much larger 
size than standard lettering in ‘‘All Capitals, 


336 SELF-TAUGHT MECHANICAL DRAWING 

either vertical or slanting, according to individual 
preference. 

The vertical lettering, Fig. 237, is constructed 
much after the same manner as the inclined style 
in regard to direction and sequence of stroke. As 
most draftsmen find it rather difficult to easily 
draw the vertical lettering (letters invariably are 
leaning forward, to the right, instead of standing 

abcolefghij kimnopqr 
stu vw xy z. 2345 6789 0 

ABCDEFGHIJKLMNOPQ 

RSTUVWXYZ 

Fig.- 237 

upright) we would advise to form the habit of 

ha\ung them lean backwards (to the left) just a 

trifle. The axis of the o in this alphabet is 

absolutely vertical; same rule applies also to 

lettys b, d, g, p, q. A similar safeguard for 

getting clear sharp corners for the upright capitals 

as illustrated in a preceding figure (235), would be 

to slightly bei^ the stems of the letters, especially 

the B, D, E, F, H, L, etc., outward. Where this 

IS neglected some very poor shapes of letters will 
result. 

Lettering on working drawings should be bold 
clear and uniform in size and the lettering should’ 
if possible, be kept clear of the drawing, that is to 


LETTERING FOR WORKING DRAWINGS 337 

say, the lettering should never be allowed to run 
into the drawing, or across it. This lettering 
should be so placed as to read from the base or 
the right hand side of a drawing. 

Dimensions should always be placed between 
centers of dimension lines, a liberal space having 
been left open for such. Where, however, the 
space for a dimension is too small, same should 
boldly be placed outside and a reference line 
from it run to the respective space. 

Lettering of titles is usually done in a somewhat 
bolder style than the ordinary descriptive matter 
on a drawing. The relative importance of portions 
of such titles should be brought out by somewhat 
larger and heavier letters, as may be required. 
In nearly every case, a title should be arranged 
symmetrically around a vertical center line. 
After the location of such a line, height and 
spacing of the different lines of letters having been 
determined the spaces equal to the width of the 
letters may be marked off with pencil on the edge 
of a strip of paper and the center of a strip placed 
on the vertical center line of the title, with its 
edge just below the line of letters to be sketched. 
The letters can then be penciled in very rapidly. 
If, after all, the spacing of a line of letters needs 
readjustment after the letters are penciled in, the 
matter may easily be rectified by working first 
to the left, then to the right of the center. 

In regard to spacing of lettering several methods 
have been advanced. The writer, however, firmly 
believes in training the draftsman’s eye to that 
end, a matter to be accomplished by placing 






338 SELF-TAUGHT MECHANICAL DRAWING 

letters composing a word as closely as possible 
against each other and liberal spaces being allowed 
between words. The optical effect of such a 
line may be studied by holding the sheet some 
distance away, when parts appearing too dark in 
a word may be relieved or lightened by spacing 
respective letters a trifle further apart. In this 
way, a sense of self-criticism will be developed and 
a habit of correctly spacing the letters formed. 

For specimens of titles, the reader is again 
referred to above-mentioned publication. 


INDEX 


A 

Accelerated motion cams, 176 
Acceleration of falling bodies, 
143 

Acme standard screw thread, 
253 

Addendum of gear teeth, 193 
Aluminum, strength of, 162 
Angle, definition of, 10 
Angle of cone clutches, 271 
Angle, to bisect an, 17 
Angles, laying out, 118 
Areas of plane figures, 92 
A. S. M. E. standard machine 
screws, 258 

Assembly drawings, 52 


B 

Back gears, 309 
Beams, cross-sections of, 156 
Beams, strength of, 159 
Belt for reversal of motion, 
crossed, 298 

Belting, horse-power of, 277 
Belting, speed of, 279 
Belting, twisted and unusual 
cases of, 282 
Belts, 276 
Belts, endless, 278 
Belts, laced, 278 
Belts, width and thickness of, 
277 

Bending, shape of parts to 
resist, 155 

Bending strength of beams, 
159 


Bevel gearing, calculating, 
230 

Bevel gears, 202 
Blue printing, 78 
Bolt heads, table of United 
States standard, 246 
Bolts, studs and screws, 243 
Bolts to withstand shock, 
248 

Brass, strength of cast, 162 
Brass wire, strength of, 158 
Broken drawings of long ob¬ 
jects, 73 

C 

Cam curve for harmonic mo¬ 
tion, 181 

Cams, comparison between 
uniform motion and accele¬ 
rated motion, 183 
Cams for high velocities, 175 
Cams, general principles, 164 
Cams with grooved edge, 172 
Cams with pivoted follower, 

167 

Cams with positive return, 
double, 173 

Cams with reciprocating mo¬ 
tion, 171 

Cams with roller follower, 

168 

Cams with straight follower, 
165 

Cams with uniform motion, 
165 

Cams with uniformly accele¬ 
rated motion, 176 
Cap screw sizes, 248 


339 


340 


INDEX 


Case for drawing instru¬ 
ments, 4 

Cast iron, strength of, 157 
Castings, stresses in, 162 
Change gears, for screw cut¬ 
ting, 302 

Check or lock nuts, 248 
Chord of circle, definition of, 
12 

Circle, area and circumfer¬ 
ence of, 92 
Circle, area of, 83 
Circle, circumference of, 80 
Circle, definition of, 11 
Circle, to find center of a, 
19 

Circles, circumscribed and in¬ 
scribed, 20 

Circles, concentric, 10 
Circles in isometric projec¬ 
tion, 48 

Circular pitch, 205 
Circular sector, area of, 93 
Circular segment, area of, 93 
Clamp coupling, 262 
Clutches, friction cone, 269 
Clutches, friction disk, 266 
Clutches, toothed, 265 
Compasses, 3 

Complement angle, definition 
of, 11 

Composition of forces, 120 
Compound idler gear, 301 
Compound gearing for screw 
cutting, 305 

Compression of machine 
parts, 154 

Compressive strength of ma¬ 
terials, 158 
Concentric circles, 10 
Cone and cylinder intersect¬ 
ing, 44 

Cone clutches, angle of, 271 
Cone clutches, friction, 269 
Cone pulleys, 239 
Cone pulleys, method of lay¬ 
ing out, 242 

Cone, surface development 
of a, 40 


Copper, strength of cast, 162 
Cosecant of an angle, 102 
Cosine of an angle, 101 
Cosines, table of, 105 
Cotangent of an angle, 102 
Cotangents, table of, 107 
Coupling, Hooke’s, 263 
Couplings, 259 
Couplings, clamp, 262 
Couplings, flange, 260 
Crank motion, quick return, 
313 

Cross-sectioning device, 7 
Cross-sections of beams, 156 
Cube, projections of a, 39 
Cube root, 82 
Cube, volume of, 94 
Cutting screw threads, gear¬ 
ing for, 302 

Cylinder and cone, intersect¬ 
ing, 44 

Cylinder, volume of, 94 
Cylinders, intersecting, 43 
Cycloid, definition of, 15 
Cycloid, to draw a, 27 
Cycloidal gear teeth, approx¬ 
imate shape of, 209 


D 

Dedendum of gear teeth, 193 
Definitions of terms, 10 
Degree, definition of, 96 
Detail drawings, 53 
Diametral pitch, 207 
Dilferential pulleys, 134 
Disk clutches, friction, 266 
Dimensions on drawings, 56 
Double cams with positive 
return, 173 

Drawings, assembly, 52 
Drawing board, 1 
Drawings, classes of lines on, 
55 

Drawings, detail, 53 
Drawings, dimensions on, 56 
Drawing instruments, 1 
Drawing paper, 8 


INDEX 


341 


Drawing pens, the use of, 7 
Drawings, sectional views on, 
66 

Drawings, working, 50 

E 

Efficiency of screws, 253 
Elevation, definition of, 33 
Ellipse, area of, 95 
Ellipse, definition of, 14 
Ellipse, to draw an, 21 
Elliptic gear quick return 
motion, 321 

Elliptic gear return motion, 
table for lay-out of, 324 
Energy and work, 146 
Energy of fly-wheel, 290 
Engines, horse-power of 
steam, 81 

Epicycloid, definition of, 15 
Epicycloidal gearing, 191 
Epicycloidal and involute 
systems of gears, compari¬ 
son between, 199 
Erasing shield, 9 

F 

Factor of safety, 151 
Falling bodies, 142 
Finishing marks on drawings, 
63 

Flange couplings, 260 
Foot-pound, definition of, 146 
Force of a blow, 147 
Forces, oblique, 124 
Forces, opposing, 125 
Forces, parallel, 123 
Forces, resultant of, 120 
Forces, resolution of, 123 
Formulas, algebraic, 79 
Formulas, transposition of, 88 
Freehand lettering for working 
dn wings, 332 

Friction cone clutches, 269 
Friction disk clutch, horse¬ 
power of, 267 


Friction disk clutches, 266 
Fulcrum, definition of, 126 
Fly-wheel, energy of, 290 
Fly-wheels for presses, 
punches, etc,, 289 
Fly-wheel, weight of, 291 


G 

Gear, compound idler, 301 
Gear, influence of the idler, 
299 

Gear quick return motion, 
elliptic, 321 

Gear teeth, approximate 
shape of, 209 

Gear teeth, laying out invo¬ 
lute, 210 

Gear teeth, Lewis’ formula 
for strength of, 218 
Gear teeth, pitch of, 205 
Gear teeth, proportions of, 
207 

Gear teeth, strength of, 213 
Gear teeth systems, compari¬ 
son between, 199 
Gear tooth, hunting, 209 
Gear tooth terms, definitions 
of, 193 

Gear, tumbler, 300 
Gearing, back, 309 
Gearing, calculating bevel, 
230 

Gearing, calculating dimen¬ 
sions of, 222 

Gearing, calculating spur, 
222 

Gearing, calculating worm, 
234 

Gearing, epicycloidal, 191 
Gearing for reversal of direc¬ 
tion of motion, 299 
Gearing for screw cutting, 
302 

Gearing, general principles 
of, 190 

Gearing, worm, 204 
Gears, bevel, 202 


342 


I 


INDEX 


Gears, interference in in¬ 
volute, 198 
Gears, 'involute, 196 
Gears, knuckle, 190 
Gears, method of drawing, 68 
Gears, proportions of, 213 
Gears, shrouded, 201 
Gears, speed ratio of, 220 
Gears, twenty degree invo¬ 
lute, 201 

Gears with radial flanks, 195 
Gears with strengthened 
flanks,’ 195 

Geometrical problems, 17 
Grooved edge cams, 172 
Guide pulleys for belts, 285 

H 

Harmonic motion cam curve, 
181 

Helix, to draw a, 47 
Heptagon, area of, 94 
Hexagon, area of, 94 
Hexagon, definition of, 14 
Hexagon, to draw a, 19 
Hoisting pulleys, 132 
Hooke’s coupling or universal 
joint, 263 
Horse-power, 149 
Horse-power of belting, 277 
Horse-power of friction cone 
clutch, 270 

Horse-power of friction disk 
clutch, 267 

Horse-power of shafting, 274 
Horse-power of steam en¬ 
gines, 81 

Hunting tooth, 209 
Hypocycloid, definition of, 15 
Hypotenuse, definition of, 98 

I 

Idler gear, compound, 300 
Idler gear, influence of the, 
299 

Inclined plane, 136 


Instrument case, 4 
Involute andepicycloidal sys¬ 
tems of gears, comparison 
between, 199 
Involute, definition of, 15 
Involute gears, 196 
Involute gears, interference 
in, 198 

Involute gear teeth, laying 
out, 210 

Involute gears, twenty de¬ 
gree, 201 

Involute rack teeth, modified 
form of, 197 
Involute, to draw an, 27 
Iron wire, strength of, 158 
Isometric projection, 48 


K 

Kirkaldy’s tests on strength 
of materials, 157 
Knuckle gears, 190 


L 

Lathe back gearing, 309 
Lathe change gears, 302 
Lathe change gears, simpli¬ 
fied rules for calculating, 
306 

Levers, 125 

Levers, compound, 128 
Lewis’ formula for strength 
of gear teeth, 218 
Line, definition of, 10 
Line, to bisect a, 17 
Lines on drawings, classes of, 
55 

Lock or check nuts, 248 

M 

Machine parts, shape of, 154 
Machine screws, 257 
Machine steel, strength of, 
158 


INDEX 


343 


Mechanics, elements of, 120 
Materials, indicating, 72 
Mechanism, trains of, 297 
Metric screw thread, form 
of, 256 

Minute, definition of, 97 
Moment, twisting or torsion¬ 
al, 272 

Motion, Newton’s laws of, 
139 

N 

Newton’s laws of motion, 139 
Nuts, check or lock, 248 
Nuts, table of United States 
standard, 246 


0 

Oblique-angled triangles, 114 
Octagon, area of, 94 
Octagon, definition of, 14 
Octagon, to draw an, 20 / 

Oldham’s coupling, 263 
Oscillation, center of, 141 

P 

Paper, drawing, 8 
Parallel forces, 123 
Parabola, definition of, 15 
Parabola, to draw a, 28 
Parallelogram, area of, 92 
Parallelogram, definition of, 
14 

Parallelogram of forces, 121 
Parallel lines, 10 
Parenthesis in formulas, 85 
Pencils, 4 
Pendulum, 141 
Pens, the use of drawing, 7 
Pentagon, area of, 93 
Pentagon, definition of, 14 
Pentagon, to draw a, 26 
Perpendicular lines, 10 
Perpendicular lines, to draw, 
18 


Pitch, circular, 205 
Pitch diameters, table of, 206 
Pitch, diametral, 207 
Plane, definition of, 10 
Plane, inclined, 136 
Point, definition of, 10 
Polygons, definition of, 14 
Positive return cams, 173 
Power transmission, screws 
for, 252 

Presses, fly-wheels for, 289 
Prism, projections of a, 34 
Prism, volume of, 94 
Projection, 32 
Projection, isometric, 48 
Pulley diameters, 281 
Pulley diameters, to calcu¬ 
late, 297 

Pulleys, cone, 239 
Pulleys, diiferential, 134 
Pulleys, guide, 285 
Pulleys, hoisting, 132 
Punches, fly-wheels for, 289 
Pyramid, surface develop¬ 
ment of a, 41 
Pyramid, volume of, 94 

Q 

Quarter-turn belting, 283 
Quick return device, Whit¬ 
worth, 319 

Quick return motions, 313 

R 

Rack teeth, modified form of 
involute, 197 

Rack with epicycloidal teeth, 
194 

Reciprocating motion cams, 
171 

Resolution of forces, 123 
Resultant of forces, 120 
Return device, Whitworth 
quick, 319 

Return motion, elliptic gear 
quick, 321 


344 


INDEX 


Return motions, quick, 313 
Reversal of direction of mo¬ 
tion, to secure, 298 
Right-angled triangles, 97 


S 

Safety, factor of, 151 
Scales, 2 

Screw cutting, gearing for, 
302 

Screw, differential, 138 
Screw, in mechanics, 138 
Screw thread, Acme stand¬ 
ard, 253 

Screw thread, form of met¬ 
ric, 256 

Screw thread, sharp V, 254 
Screw thread, Whitworth, 
255 

Screw threads, drawing, 74 
Screw threads, table of 
United States standard, 246 
Screw threads. United States 
standard, 245 

Screw threads, wrench action 
on, 249 

Screws, bolts and studs, 243 
Screws, dimensioning, 62 
Screws, efficiency of, 253 
Screws for power transmis¬ 
sion, 252 

Screws, machine, 257 
Screws, set, 256 
Screws, square threaded, 251 
Secant of an angle, 102 
Second, definition of, 97 
Sections on drawings, 66 
Set-screws, 256 
Shade lines, 77 
Shafting, horse-power of, 274 
Shafts, 272 

Shafts at right angles, belt¬ 
ing between, 283 
Shafts, Thurston’s rule for 
strength of, 220 
Shapers, quick return mo¬ 
tion for, 316 


Sharp V-thread, 254 
Shearing strength of mate¬ 
rials, 240 

Shearing strength of shaft¬ 
ing, torsional, 273 
Shears, fly-wheels for power, 
289 

Shrouded gears, 201 
Sine of an angle, 101 
Sines, table of, 104 
Solid, definition of, 10 
Speed of belting, 279 
Speed ratio of gears, 220 
Speed ratio of sprocket 
wheels, 189 

Speed, to secure increase of, 
297 

Sphere, area and volume of, 
94 

Spherical sector, volume of, 
94 

Spherical segment, volume 
of, 95 

Spiral, to draw a, 26 
Sprocket wheels, 185 
Sprocket wheels, graphical 
method of laying out, 187 
Sprocket wheels, speed ratio 
of, 189 

Spur gearing, calculating, 
222 

Spur gears, method of draw- 
Ing, 68 

Square root, 82 

Square threaded screws, 251 

Steel castings, strength of, 

157 

Steel, strength of machine, 

158 

Steel, strength of structural, 
162 

Steel wire, strength of, 158 
Stepped cone pulleys, 239 
Strength of gear teeth, 213 
Strength of gear teeth, 
Lewis’ formula for, 218 
Strength of materials, 151 
Strength of”:materials, Kirk- 
aldy’s tests on, 157 


INDEX 345 


Strength of materials, shear¬ 
ing, 260 

Strength of shafting, tor¬ 
sional shearing, 273 
Strength of shafts, twisting, 
272 

Stresses in castings, 162 
Studs, screws and bolts, 243 
Supolement angle, definition 
of, 11 

Surface, definition of, 10 


T 

Tangent, definition of, 13 
Tangent of an angle, 101 
Tangent to a circle, to draw 
a, 19 

Tangents, table of, 106 
Technique of mechanical draft¬ 
ing, 326 

Tension in belts, 276 
Tension, machine parts sub¬ 
jected to, 154 
Thickness of belts, 277 
Thread, Acme standard 
screw, 253 

Thread cutting, gearing for, 
302 

Thread, form of metric screw, 
256 

Thread, sharp V, 254 
Thread, Whitworth screw, 
255 

Thread, drawing screw, 74 
Threads, screws with square, 
251 

Threads, United States 
Standard screw, 245 
Thurston’s mle for strength 
of shafts, 220 
Toothed clutches, 265 
Torsional strength of shafts, 
272 

Trains of mechanism, 297 


Transposition of formulas, 88 
Triangle, area of, 91 
Triangles, solution of, 96 
Trigonometry, elements of, 
96 

Tumbler gear, 300 
Twisting strength of shafts, 
272 

u 

Uniform motion cams, 165 
Uniformly accelerated mo¬ 
tion cams, 176 

United States standard screw 
thread, 245 
Universal joint, 263 


V 

V-Thread, sharp, 254 
Vertex of angle, definition 
of, 10 

Views on working drawings, 
number of, 50 
Volume of solids, 94 


w 

Weight of fly-wheel, 291 
Whitworth quick return de¬ 
vice, 319 

Whitworth screw thread, 255 
Width of belts, 277 
Wire, strength of, 158 
Work and energy, 146 
Working drawings, 50 
Worm gearing, 204 
Worm gearing, calculating, 
234 

Wrench action on screw 
threads, 249 

Wrought iron, strength of, 
157 


\ 

I 

j 

ft. 

'A 

i 



LATEST 

REVISED 

CATALOGUE 

of the Best 

Practical and Mechanical Books 

Including Automobile and Aviation Books 


PRACTICAL BOOKS FOR PRACTICAL MEN 

Each Book in this Catalogue is written by an 
Expert and is written so you can understand it 

PUBLISHED BY 

The Norman W. Henley Publishing Co. 

2 West 45th Street, New York, U. S. A. 

Established 1890 

Any Book in this Catalogue sent prepaid on receipt of price 
Manuscripts solicited on Practical Subjects 












INDEX 


PAGE 

/lorasivcs and Abrasive Wheels. 29 

Accidents. 25 

Air Brakes.24, 26 

Arithmetic.13, 27, 37 

Automobile Books.3, 4, 5, 6 

Automobile Charts.6,7 

Automobile Ignition Systems. 7 

Automobile Lighting. 4 

Automobile Questions and Answers.... 5 

Automobile Repairing. 5 

Automobile Starting Systems. 4 

Automobile Trouble Charts.6, 7 

Automobile Welding. 6 

Aviation. 8 

Bells, Electric. 14 

Bevel Gear. 21 

Boats, Motor. 30 

Boiler Room Chart. 9 

Boilers, Marine. 30 

Brazing. 9 

Cams. 21 

Carburetion Trouble Charts. 7 

Carburetors. 4 

Car Charts. 9 

Cements. 12 

Change Gear. 21 

Charts.6, 7, 9 

Chemistry. 10 

Coal. 24 

Combustion. 19 

Concrete.10, 11, 12 

Concrete for Farm Use. 11 

Concrete for Shop Use. 11 

Cosmetics. 32 


Dictionary.12, 17 

Dies. 12 

Drawing.13, 32 

Drawing for Plumbers. 32 

Dynamo Building. 14 

Electric Bells. 14 

Electric Dictionary. 17 

Electric Switchboards. 15 

Electric Toy Making. 16 

Electric Wiring. 15 

Electricity.13, 14, 15, 16, 17, 18 

Electroplating. 19 

Encyclopedia. 29 

Engine, Aviation. 8 

E-T Air Brake. 26 

Factory Management. 19 

Ford Automobile. 4 

Ford Tractor. 4 

Ford Trouble Chart. 7 

Formulas and Recipes. 34 

Fuel. 19 

Gas Engines.20, 21 

Gas Tractor. 39 

Gearing and Cams. 21 

Glossary Aviation Terms. 8 

Heating. 37 

High Frequency Apparatus. 15 

Horse-Power Ciiart. 36 

Hot Water Heating. 37 

House Wiring.17, 18 

Hydraulics. 22 

Ice. 22 

Ignition Systems. 4 

Ignition Trouble Chart. 7 

India Rubber. 35 

Interchangeable Manufacturing. 27 

Im^entions. 22 

Kerosene Carburetors. 4 

Knots. 23 

Any of these hooks promptly sent prepaid to 

HOW TO liEMIT.—By Postal Tvlontiv Order. Express Money Order. Hank Draft* or Registered Letter 


PAGl 

Lathe Work. 2;| 

Link Motions. 2‘i 

Liquid Air. 2A 

Locomotive Boilers. 21 

Locomotive Breakdowns. 21 

Locomotive Engineering.24, 25, 2( 

Machinist Books.27, 28, 2j 

Manual Training. 3( 

Marine Engineering. 3( 

Marine GasoUne Engines. 2. 

Mechanical Drawing. i; 

Mechanical Movements. 2\‘ 

Metal Work. 11 

Model Making. 2{ 

Motor Boats. 3( 

Motorcycles.7, 3] 

Motor Truck. 1 

Naval Engineering. 3 (| 

Patents. 21 

Pattern Making. 31 

Perfumery. 31 

Perspective. IJ; 

Plumbing.32, 3fi 

Punches. 11 

Producer Gas. 21 

Questions and Answers on Automobile. f 

Questions on Heating. 3S 

Radio Time Signal Receiver.16, 11 

Railroad Accidents. 2£ 

Railroad Charts. *..... £ 

Recipe Book. 3-^ 

Refrigeration. 21 

Repairing Automobiles. f 

Rope Work. 2c 

Rubber. 

Rubber Stamps. 3^ 

Saw Filing. 3f 

Saws, Management of. 3£ 

Screw Cutting. 3f 

Shipbuilders’ Pocket Book. 3C 

Shop Construction.. 27 

Shop Management. 2? 

Shop Practice.27, 28, 2g' 

Shop Tools. 2£ 

Sketching Paper. 13 

Slide Valve. 24 

Soldering. 9 

Splices and Rope Work. 23 

Steam Engineering.35, 36, 37 

Steam Heating. 37 

Steel. 38 

Storage Batteries. 18 

Submarine Chart. 9 

Switchboards. 15 

Tapers. 23: 

Telegraphy, Wireless .14, 16, 18' 

Telephone. 18; 

Thread-Cutting. 27; 

Tool Making. 27 

Tool Steel. 38j 

Toy Making. 16 

Tractive Power Chart. 9. 

Tractor, Gas. 391 

Train Rules. 26 

Vacuum Heating. 38 

Valve Setting. 24 

Ventilation. 37; 

Walschaert Valve Gear. 26; 

W ater proofing. 12! 

Welding.•..6, 39i 

Wireless Telegraphy.14, 16, 18 

Wiring.15, 17, 181 

Wiring Diagrams. 15 

any address in the world on receipt of price. 

























































































































































CATALOGUE OF GOOD, PRACTICAL BOOKS 


AUTOMOBILES 


THE MODERN GASOLINE AUTOMOBILE—ITS DESIGN, CONSTRUC¬ 
TION, MAINTENANCE AND REPAIR By Victor W. Page, M.E. 

The latest and most complete treatise on the Gasoline Automobile ever issued. Written 
in simple language by a recognized authority, familiar with every branch of the auto¬ 
mobile industry. Free from technical terms. Everything is explained so simply 
that anyone of average inteUigence may gain a comprehensive knowledge of the 
gasoUne automobile. The information is up-to-date and includes, in addition to an 
exposition of principles of construction and description of all types of automobiles and 
their components, valuable money-saving hints on the care and operation of motor¬ 
cars propelled by internal combustion engines. Among some of the subjects treated 
might be mentioned: Torpedo and other symmetrical body forms designed to reduce 
air resistance; sleeve valve, rotary valve and other types of silent motors: increasing 
tendency to favor worm-gear power-transmission: universal application of magneto 
ignition; development of automobile electric-lighting systems: block motors; under¬ 
slung chassis: application of practical self-starters; long stroke and offset cylinder 
motors; latest automatic lubrication systems; silent chains for valve operation and 
change-speed gearing; the use of front wheel brakes and many other detail refinements. 
By a careful study of the pages of this book one can gain practical knowledge of auto¬ 
mobile construction that will save time, money and worry. The book tells you just 
what to do, how and when to do it. Nothing has been omitted, no detail has been 
slighted. Every part of the automobile, its equipment, accessories, tools, supplies, 
spare parts necessary, etc., have been discussed comprehensively. If you are or 
intend to become a motorist, or are in any way interested in the modern Gasoline 
Automobile, this is a book you cannot afford to be without. 1032 pages— and 
more than 1,000 new and specially made detail illustrations, as well as many full-page 
and double-page plates, showing all parts of the automobile. Including 12 large 
folding plates. New Edition. Price .$4.00 

WHAT IS SAID OP THIS BOOK: 

“It is the best book on the Automobile seen up to date.”—J. H. Pile, Associate Editor 
Automobile Trade Journal. 

“Every Automobile Owner has use for a book of this character.”— The Tradesman. 
“This book is superior to any treatise heretofore published on the subject.”— The 
Inventive Age. 

“We know of no other volume that is so complete in all its departments, and in which 
the wide field of automobile construction with its mechanical intricacies is so plainly 
handled, both in the text and in the matter of illustrations.”— The Motorist. 

“The book is very thorough, a careful examination failing to disclose any point in 
connection with the automobile, its care and repair, to have been overlooked.”— 
Iron Age. 

“Mr. Page has done a great work, and benefit to the Automobile Field.”—W. O. 
Hasford, Mgr. Y. M. C. A. Automobile School, Boston, Mass. 

“It is just the kind of a book a motorist needs if he wants to understand his car.”— 
American Thresherman. 

THE MODERN MOTOR TRUCK, ITS DESIGN, CONSTRUCTION, OPERA¬ 
TION AND REPAIR. By Victor W. Page. 

Just off the press and treats on all types of motor trucks and industrial tractors and 
trailers. It considers all types of trucks, gasoline and electric, and all varieties of 
truck bodies. This book is written in language everyone can understand and is 
not in any .sense of the word a technical treatise. It is a practical volume that will 
make special appeal to the truck driver who seeks to better his position and to the 
mechanic charged with the repair and upkeep of trucks. The factory or business 
executive who wants to obtain a complete working knowledge of truck operation 
problems will And this book a reference work of great value. The truck salesman or 
automobile dealer will find that this work contains information that means money 
to them. All garage and service station men should have a copy of this book for 
reference because truck construction differs from passenger car design in many im¬ 
portant respects. Anyone who reads this book is in touch with all the practical 
features that have been tested out in real service. 1921 Edition. Cloth, 6x9, 
962 pages, 750 illustrations Price.$5.0Q 


5 








CATALOGUE OF GOOD, PRACTICAL BOOKS 


THE MODEL T FORD CAR, ITS CONSTRUCTION, OPERATION AND 
REPAIR, INCLUDING THE FORDSON FARM TRACTOR, F. A. LIGHT¬ 
ING AND STARTING SYSTEM, FORD MOTOR TRUCK. By Victor 
W. Page. 

This is the most complete and practical instruction book ever published on the Ford 
car and Fordson tractor. All parts of the Ford Model T car and Fordson tractor 
are described and illustrated in a comprehensive manner. The construction is 
fully treated and operating principle made clear to everyone. Complete instructions 
for driving and repairing are given. To the New Revised Edition matter has been 
added on the Ford Truck and Tractor Conversion Sets and Genuine Fordson Tractor. 
All parts are described. All repair proces.ses illustrated and fully explained. Written 
so all can understand—no theory, no guesswork. New revised and enlarged Edition 
just pubUshed. 153 illustrations, 410 pages, 2 large folding plates. Price . $2.00 

AUTOMOBILE STARTING, LIGHTING AND IGNITION SYSTEMS. By 

Victor W. Page, M.E. 

This practical volume has been written with special reference to the requirements of the 
non-technical reader desiring easily understood, explanatory matter, relating to all 
types of automobile ignition, starting and lighting systems. It can be understood by 
anyone, even without electrical knowledge, because elementary electrical principles are 
considered before any attempt is made to discuss features of the various systems. 
These basic principles are clearly stated and illustrated with simple diagrams. All the 
leading systems of starting, lighting and ignition have been described and illustrated with 
the co-operation of the experts employed by the manufacturers. Wiring diagrams are 
shown in both technical and non-technical forms. All symbols are fully explained. It 
is a comprehensive review of modern starting and ignition system practice, and includes 
a complete exposition of storage battery construction, care and repair. All types of 
starting motors, generators, magnetos, and all ignition or lighting system units are 
fully explained. The systems of cars already in use as well as those that are to come 
are considered. Every person in the automobile business needs this volume. 

Cloth. 815 pages, 492 illustrations, 3 folding plates. New revised and enlarged 
edition. Price.•. $3.00 


GASOLINE AND KEROSENE CARBURETORS, CONSTRUCTION, IN¬ 
STALLATION AND ADJUSTMENT. By Victor W. Page. A new up- 
to-date Book on Modern Carburetion Practice. 


This is a simple, comprehensive, and authoritative treatise for practical men ex¬ 
plaining all basic principles pertaining to carburetion, sliowing how Uquid fuels are 
vaporized and turned into gas for operating all types of internal combustion engines in¬ 
tended to operate on vapors of gasoline, kerosene, benzol, and alcohol. All leading types 
of carburetors are described in detail, special attention being given to the forms devised 
to use the cheaper fuels such as kerosene. Carburetion troubles, fuel system troubles, 
carburetor repairs and installation, electric primers and economizers, hot spot mani¬ 
folds and all modern carburetor developments are considered in a thorough manner. 
Methods of adjusting all types of carburetors are fully discussed as well as sugges¬ 
tions for securing maximum fuel economy and obtaining highest engine power. 

This book is invaluable to repairmen, students, and motorists, as it includes the 
most complete exposition on kerosene carburetors ever published. The drawings 
showing carburetor construction are made from accurate engineering designs and 
show all parts of late types of carburetors. 213 pages. 89 illustrations. . $2.00 


HINTS AND TIPS FOR AUTOMOBILISTS. By Victor W. Page. 

The book is ideal for the busy man or woman who wants to know about car operation 
and upkeep because of the economies possible when an automobile is intelligently 
operated. It contains many money-saving hints and a brief simple exposition of 
location and remedy of roadside troubles apt to occur under ordinary operating 
conditions. Price. 75 cents 














CATALOGUE OF GOOD, PRACTICAL BOOKS 


AUTOMOBILE REPAIRING MADE EASY. By Victor W. Page, M.E. 

A comprehensive, practical exposition of every phase of modern automobile repairing 
practice. Outlines every process incidental to motor car restoration. Gives plans for 
workshop construction, suggestions for equipment, power needed, machinery and tools 
necessary to carry on the business successfully. Tells how to overhaul and repair all 
parts of all automobiles. Everything is explained so simply that motorists and students 
can acquire a full working knowledge of automobile repairing. This work starts wit h 
the engine, then considers carburetion, ignition, cooling and lubrication systems. The 
clutch, chang'^-speed gearing and transmission system are considered in detail. Contain.? 
instructions for repairing all types of axles, steering gears and other chassis parts. 
Many tables, short cuts in figuring and rules of practice are given for the mechanic. 
Explains fully valve and magneto timing, “tuning” engines, systematic location of 
trouble, repair of ball and roller bearings, shop kinks, first aid to injured and a multi¬ 
tude of subjects of interest to all in the garage and repair business. 

This book contains special instructions on electric starting, lighting and ignition systems, 
tire repairing and rebuilding, autogenous welding, brazing and soldering, heat treatment of 
steel, latest timing practice, eight and twelve-cylinder motors, etc. 55^x8. Cloth. 1060 
pages, 1,000 illustrations, 11 folding plates. Price.$4.00 

WHAT IS SAID OF THIS BOOK: 

“ ‘Automobile Repairing Made Easy’ is the best book on the subject I have ever seen 
and the only book I ever saw that is of any value in a garage.”—Fred Jeffrey, Martins- 
burg. Neb. 

“I wish to thank you for sending me a copy of ‘Automobile Repairing Made Easy.’ I 
do not think it could be excelled.”—S. W. Gisriel, Director of Instruction, Y. M. C. A., 
Philadelphia, Pa. 


QUESTIONS AND ANSWERS RELATING TO MODERN AUTOMOBILE 
CONSTRUCTION, DRIVING AND REPAIR. By Victor W. Page, M.E. 

A practical self-instructor for students, mechanics and motorists, consisting of thirty- 
seven lessons in the form of questions and answers, written with special reference to the 
requirements of the non-technical reader desiring easily understood, explanatory 
matter relating to all branches of autoinobiling. The subject-matter is absolutely 
correct and explained in simple language. If you can’t answer all of the following 
questions, you need this work. The answers to these and nearly 2000 more are to 
be found in its pages. Give the name of all important parts of an automobile and 
describe their functions? Describe action of latest types of kerosene carburetors? 
What is the difference between a “double” ignition system and a “dual” ignition 
system? Name parts of an induction coil? How are valves timed? What is an 
electric motor starter and how does it work? What are advantages of worm drive 
gearing? Name all important types of ball and roller bearings? What is a “three- 
quarter” floating axle? What is a two-speed axle? What is the Vulcan electric gear 
shift? Name the causes of lost power in automobiles? Describe all noises due to 
deranged mechanism and give causes? How can you adjust a carburetor by the 
color of the exhaust gases? What causes “popping” in the carburetor? What tools 
and supplies are needed to equip a car? How do you drive various makes of cars? 
What is a differential lock and where is it used? Name different systems of wire 
wheel construction, etc., etc. A popular work at a popular price. 5J4x7H. Cloth. 
701 pages, 387 illustra,tions, 3 folding plates. New revi.sed edition. Price $2.60 

WHAT IS SAID OF THIS BOOK: 

“If you own a car—^get this book .”—The Glassworker. 

“Mr. Page has the faculty of making difficult subjects plain and understandable.”— 
Bristol Press. 

“We can name no writer better qualified to prepare a book of instruction on auto¬ 
mobiles than Mr. Victor W. Page .”—Scientific American. 

“The best automobile catechism that has appeared .”—Automobile Topics. 

“There are few men, even with long experience, who will not find this book useful. 
Great pains have been taken to make it accurate. Special recommendation must be 
given to the illustrations, which have been made specially for the work. Such ex¬ 
cellent books as this greatly assist in fully understanding your automobile.”—En- 
gineering News. 


5 






CATALOGUE OF GOOD, PRACTICAL BOOKS 


HOW TO RUN AN AUTOMOBILE. By Victor W. Page. 

This treatise gives concise instructions for starting and running all makes of gasoline 
automobiles, how to care for them, and gives distinctive features of control. De¬ 
scribes every step for shifting gears, controlling engine, etc. Among the chapters 
contained are: I. Automobile Parts and Their Functions. II. General Starting 
and Driving Instructions. III. Control Systems—Care of Automobiles. Thoroughly 
illustrated. 178 pages. 72 illustrations. Price.$1.50 

THE AUTOMOBILIST’S POCKET COMPANION AND EXPENSE RECORD. 

By Victor W. Page. 

This book is not only valuable as a convenient cost record, but contains much in¬ 
formation of value to motorists. Includes a condensed digest of auto laws of all 
States, a lubrication schedule, hints for care of storage battery, and care of tires, 
location of road troubles, anti-freezing solutions, horse-power table, driving hints 
and many useful tables and recipes of interest to all motorists. Not a technical 
book in any sense of the word, just a collection of practical facts in simple language 
for the every-day motorist. Convenient pocket size. Price.$1.50 


AUTOMOBILE WELDING WITH THE OXY-ACETYLENE FLAME. By 

M. Keith Dunham. 

Explains in a simple manner apparatus to be used, its care, and how to construct 
necessary shop equipment. Proceeds then to the actual welding of all automobile 
parts, in a manner understandable by everyone. Gives principles never to be for¬ 
gotten. This book is of utmost value, since the perplexing problems arising when 
metal is heated to a melting point are fully explained and the proper methods to 
overcome them shown. 167 pages. Fully illustrated. Price. $1.50 


AUTOMOBILE, AVIATION AND MOTORCYCLE CHARTS 


AVIATION CHART—LOCATION OF AIRPLANE POWER PLANT TROUBLES 
MADE EASY. By Major Victor W. Page, A.S., S.C.U.S.R. 

A large chart outlining all parts of a typical airplane power plant, showing the points 
where trouble is apt to occur and suggesting remedies for the common defects. In¬ 
tended especially for aviators and aviation iRechanics on school and field duty. 
Price. 35 cents 


CHART. GASOLINE ENGINE TROUBLES MADE EASY—A CHART SHOW¬ 
ING SECTIONAL VIEW OF GASOLINE ENGINE. Compiled by Victor 
W. Page, M.E. 

It shows clearly all parts of a typical four-cylinder gasoline engine of the four-cycle 
type. It outlines distinctly all parts liable to give trouble and also details the de¬ 
rangements apt to interfere with smooth engine operation. 

Valuable to students, motorists, mechanics, repairmen, garagemen, automobile sales¬ 
men, chauffeurs, motorboat owners, motor-truck and tractor drivers, aviators, motor¬ 
cyclists, and all others who have to do with gasoline power plants. 

It simpUftes location of all engine troubles, and while it will prove invaluable to the 
novice, it can be used to advantage by the more expert. It should be on the walls of 
every public and private garage, automobile repair shop, clubhou.se or school. It can 
be carried in the aiitomobile or pocket with ease, and will insure against loss of time 
when engine trouble manifests itself. 

This sectional view of engine is a complete review of all motor troubles. It is prepared 
by a practical motorist for all who motor. More information for the money than ever 
before offered. No details omitted. Size 25x38 inches. Securely mailed on receipt 

of.. cents 

6 









CATALOGUE OF GOOD, PRACTICAL BOOKS 


CHART. LOCATION OF FORD ENGINE TROUBLES MADE EASY. Com- 

piled by Victor W. Page, M.E. 

This shows clear sectional views depicting all portions of the Ford power plant and 
auxiliary groups. It outlines clearly all parts of the engine, fuel supply system, igni- 
tion group and cooling system, that are apt to give trouble, detailing all derangements 
that are liable to make an engine lose power, start hard or work irregularly. This 
chart is valuable to students, owners, and drivers as it simpliucs location of all engine 
faults. Of great advantage as an instructor for the novice, it can be used equally well 
by the more expert as a work of reference and review. It can be carried in the tool¬ 
box or pocket with ea.se and will save its cost in labor eliminated the first time engine 
trouble manifests itself. Prepared with special reference to the average man's needs 
and is a practical review of all motor troubles because it is based on the actual ex¬ 
perience of an automobile engineer-mechanic with the mechanism the chart describes. 
It enables the non-technical owner or operator of a Ford car to locate engine de¬ 
rangements by systematic search, guided by easily recognized symptoms instead of by 
guesswork. It makes the average owner independent of the roadside repair shop 
when touring. Must be seen to be appreciated. Size 25x:i8 inches. Printed on 
heavy bond paper. Price. 35 cents 

CHART. LUBRICATION OF THE MOTOR CAR CHASSIS. Compiled by 
Victor W. Page, M.E. 

This chart presents the plan view of a typical six-cylinder chassis of standard design 
and all parts are clearly indicated that demand oil, also the freciuency with which they 
must be lubricated and the kind of oil to use. A practical chart for all interested in 
motor-car maintenance. Size 24x38 inches. Price. 35 cents 

CHART. LOCATION OF CARBURETION TROUBLES MADE EASY. Com¬ 
piled by Victor W. Page, M.E. 

This chart shows all parts of a typical pressure feed fuel supply system and gives 
causes of trouble, how to locate defects and means of remedying them. Size 24x38 
inches. Price.35 cents 

CHART. LOCATION OF IGNITION SYSTEM TROUBLES MADE EASY. 
Compiled by Victor W. Page, M.E. 

In this diagram all parts of a typical double ignition system using battery and magneto 
current are shown, and suggestions are given for readily finding ignition troubles and 
eliminating them when found. Size 24x38 inches. Price.35 cents 

CHART. LOCATION OF COOLING AND LUBRICATION SYSTEM FAULTS. 

Compiled by Victor W. Page, M.E. 

This composite diagram shows a typical automobile power plant using pump circulated 
water-cooling system and the most popular lubrication method. Gives suggestions 
for curing all overheating and loss of power faults due to faulty action of the oiling 
or cooling group. Size 24x38 inches. Price.35 cents 

CHART. LOCATION OF STARTING AND LIGHTING SYSTEM FAULTS. 

The most complete chart yet devised, showing all parts of the modern automobile 
starting, lighting and ignition systems, giving instructions for systematic location of 
all faults in wiring, lamps, motor or generator, switches and all other units. Invalu¬ 
able to motorists, chauffeurs and repairmen. Size 24x38 inches. Price . 35 cents 

CHART. MOTORCYCLE TROUBLES MADE EASY. Compiled by Victor 
W. Page, M.E. 

A chart showing sectional view of a single-cylinder gasoline engine. This chart 
simplifies location of all power-plant troubles. A single-cylinder motor is shown for 
simplicity. It outlines distinctly all parts liable to give trouble and also details the 
derangements apt to interfere with smooth engine operation. This chart will prove 
of value to all who have to do with the operation, repair or sale of motorcycles. No 
details omitted. Size 30x20 inches. Price.35 cents 

7 












CATALOGUE OF GOOD, PRACTICAL BOOKS 


AVIATION 


A B C OF AVIATION. By Major Victor W. Faq^. 

This book describes the basic principles of aviation, tells how a balloon or dirigible 
is made and why it floats in the air. Describes how an airplane flies. It shows in 
detail the different parts of an airplane, what they are and what they do. Describes 
all types of airplanes and how they differ in construction: as well as detailing the 
advantages and disadvantages of different types of aircraft. It includes a complete 
dictionary of aviation terms and clear drawings of leading airplanes. The reader 
will And simple instructions for unpacking, setting up, and rigging airplanes. A 
full description of airplane control principles is given and methods of flying are dis¬ 
cussed at length. 

Tills book answers every question one can ask about modern aircraft, their con¬ 
struction and operation. A self-educator on aviation without an equal. 274 pages. 
130 specially made illustrations with 7 plates. Price.$2.50 


AVIATION ENGINES—DESIGN; CONSTRUCTION; REPAIR. By Major 
Victor W. Page, A.S., S.C.U.S.R. 

This treatise, written by a recognized authority on all of the practical aspects of 
internal combustion engine construction, maintenance, and repair, fills the need as 
no other book does. The matter is logically arranged; all descriptive matter is 
simply expressed and copiously illustrated, so that anyone can understand airplane 
engine operation and repair even if without previous mechanical training. This 
work is invaluable for anyone desiring to become an aviator or aviation mechanic. 
The latest rotary types, such as the Gnome Ivlonosoupape, and LeRhone, are fully 
explained, as well as the recently developed Vee and radial types. The subjects 
of carburetion, ignition, cooling, and lubrication also are covered in a thorough manner. 
The chapters on repair and maintenance are distinctive and found in no other book 
on tills subject. Not a technical book, but a practical, easily understood work of 
reference for all interested in aeronautical science. 589 pages. 253 illustrations. 
Price, net..$3.00 

GLOSSARY OF AVIATION TERMS —ENGLISH-FRENCH; FRENCH- 
ENGLISH. By Major Victor W. Page, A.S., S.C.U.S.R., and Lieut. 
Paul Montariol, of the French Flying Corps. 

A complete glossary of practically all terms used in aviation, having lists in both 
French and English with equivalents in either language. Price, net . . $1.00 


APPLIED AERODYNAMICS. By G. P. Thompson. ' 

This is a scientific and mathematical treatise that has a special appeal to the student 
and engineer who are seeking exact information on the aerodynamics of heavier-than- 
air craft and data on airplane design testing. This book gives an up-to-date presen¬ 
tation of the existing state of Aeronautical Science. In addition to'a very full dii. 
cussion of the qualities which determine the speed and rate of climb of an aeroplane 
and the method by which they can be calculated, special attention is paid to stability 
•—a problem now fairly well understood, and to controllability—our knowledge of 
which is at present in a much more elementary state. A.ttention is directed to the 
numerous directions in which further information is required, especially in the form of 
full-scale experiments. 312 pages (7 x 10). Illustrated W'ith over 142 Diagrams 
and Graphic Charts. Price. $12.50 

AVIATION CHART—LOCATION OF AIRPLANE POWER PLANT TROUBLES 
MADE EASY. By Major Victor W. Page, A.S., S.C.U.S.R. 

A large chart outlining all parts of a typical airplane power plant, showing the points 
where trouble is apt to occur and suggesting remedies for the common defects In¬ 
tended especially for aviators and aviation mechanics on school and field duty 
I*«ce.. cent* 


8 











CATALOGUE OF GOOD, PRACTICAL BOOKS 


BRAZING AND SOLDERING 


BRAZING AND SOLDERING. By James F. Hobart. 

The only book that shows you just how to handle any job of brazing or soldering that 
comes along; it tells you what mixture to use, how to make a furnace if you need one. 
Full of valuable kinks. The fifth edition of this book has just been published, and to 
it much new matter and a large number of tested formulas for all kinds of solders and 
fluxes have been added. Illustrated. 35 cents 

SOLDERING AND BRAZING. By Raymond Francis Yates. 

This treatise gives all the necessary “kinks” that will enable one to accomplish suc¬ 
cessful soldering. If a mechanic has not succeeded in his soldering, this book may 
tell him just what be needs to produce good work—something that he may hereto¬ 
fore have forgotten. Price. 75 cents 


CHARTS 


MODERN SUBMARINE CHART. 

A cross-section view, showing clearly and distinctly all the interior of a Submarine of 
the latest type. You get more information from this chart, about the construction and 
operation of a Submarine, than in any other way. No details omitted—everything 
is accurate and to .scale. All the machinery and devices fitted in a modern Submarine 


Boat are shown.35 cents 

BOX CAR CHART. 

A chart showing the anatomy of a box car, having every part of the car numbered and 
its proper name given in a reference list.35 cent.*^ 

GONDOLA CAR CHART. 

A chart showing the anatomy of a gondola car, having every part of the car numbered 
and its proper reference name given in a reference list. 35 cents 

PASSENGER-CAR CHART. 

A chart showing the anatomy of a passenger-car, having every part of the car numbered 
and its proper name given in a reference list. 35 cents 

STEEL HOPPER BOTTOM COAL CAR. 

A chart showing the anatomy of a steel Hopper Bottom Coal Car, having every part 
of the car numbered and its proper name given in a reference list. 35 cent.s 

TRACTIVE POWER CHART. 


A chart whereby you can find the tractive power or drawbar pull of any locomotive 
without making a figure. Shows what cylinders are equal, how driving wheels and 
steam pressure affect the power. What sized engine you need to exert a given drawbar 
pull or anything you desire in this line. 50 cents 

HORSE-POWER CHART 

Shows the horse-power of any stationary engine without calculation. No matter what 
the cylinder diameter of stroke, the steam pressure of cut-off, the revolutions, or 
whether cn-jdensing or non-condensing, it’s all there. Easy to use, accurate, and 
saves time and o^.^ulations. Especially useful to engineers and designers, 50 cents 

BOILER ROOM CHART. By Geo. L. Fowler. 

chart_size 14x28 inches—showing in isometric perspective the mechanisms be¬ 

longing in a modern boiler room. The various parts are shown broken or removed, 
so that the internal construction is fully illustrated. Each part is given a reference 
number, and these, with the corresponding name, are given in a glossary printed at 
the sides... 85 cents 


9 
















CATALOGUE OF GOOD, PRACTICAL BOOKS 


CHEMISTRY 


HOW TO MAKE AND USE A SMALL CHEMICAL LABORATORY. By 

Raymond Francis Yates. 

The treatise covers all of the essentials of elementary chemistry. The law ©f definite 
proportions, solutions, crystalloids, colloids, electrolysis, etc., are explained. The 
second part of the book is devoted to chemical and electro-chemical experiments. 
Only those experiments that will tend to broaden the reader’s knowledge of chemistry 
in general have been chosen. Price.75 cents 


CONCRETE 


JUST PUBLISHED—CONCRETE WORKERS’ REFERENCE BOOKS. A 
SERIES OF POPULAR HANDBOOKS FOR CONCRETE USERS. 

Prepared by A. A. Houghton .Each 75 cents 

The author, in preparing this Series, has not only treated on the usual types of construction, 
but explains and illustrates molds and systems that are not patented, but which are equal 
in value and often superior to those restricted by patents. These molds are very easily and 
cheaply constructed and embody simplicity, rapidity of operation, and the most successful 
results in the molded concrete. Each of these Twelve books is fully illustrated, and the 
subjects are exhaustively treated in plain English. 

CONCRETE WALL FORMS. By A. A. Houghton. 

A new automatic wall clamp is illustrated with working drawings. Other types ol 
wall forms, clamps, separators, etc., are also illustrated and explained. 

(No. 1 of Series). 75 cents 

CONCRETE FLOORS AND SIDEWALKS. By A. A. Houghton. 

The molds for molding squares, hexagonal and many other styles of mosaic floor and 
sidewalk blocks are fully illustrated and explained. (No. 2 of Series) . . 75 cents 

PRACTICAL CONCRETE SILO CONSTRUCTION. By A. A. Houghton. 

Complete working drawings and specifications are given for several styles of concrete 
silos, with illustrations of molds for monolithic and block silos. The tables, data, and 
information presented in this book are of the utmost value in planning and constructing 
all forms of concrete silos. (No. 3 of Series) . . . . '. 75 cents 

MOLDING CONCRETE CHIMNEYS, SLATE AND ROOF TILES. By A. A. 

Houghton. 

The manufacture of all types of concrete slate and roof tile is fully treated. Valuable 
data on all forms of reinforced concrete roofs are contained within its pages. The 
construction of concrete chimneys by block and monolithic systems is fully illustrated 
and described. A number of ornamental designs of chimney construction with molds 
are shown in this valuable treatise. (No. 4 of Series.) ....... 75 cents 

MOLDING AND CURING ORNAMENTAL CONCRETE. By A. A. Houghton. 

The proper proportions of cement and aggregates for various finishes, also the method 
of thoroughly mixing and placing in the molds, are fully treated. An exhaustive 
treatise on this subject that every concrete worker will find of daily use and value 
(No. 5 of Series.). 75 cents 

CONCRETE MONUMENTS, MAUS0LEUMS;;AND BURIAL VAULTS. By 

A. A. Houghton. 

The molding of concrete monuments to imitate the most expensive cut stone is ex¬ 
plained in this treatise, with working drawings of easily built molds. Cutting in¬ 
scriptions and designs are also fully treated. (No. 6 of Series.) ... 75 cents 

10 














CATALOGUE OF GOOD, PRACTICAL BOOKS 


MOLDING CONCRETE BATHTUBS, AQUARIUMS AND NATATORIUMS. 

By A. A. Houghton. 

Simple molds and instruction are given for molding many styles of concrete oathtubs, 
swimming-pools, etc. These molds are easily built and permit rapid and successful 
work. (No. 7 of Series.). 75 cents 

CONCRETE BRIDGES, CULVERTS AND SEWERS. By A. A. Houghton. 

A number of ornamental concrete bridges with illustrations of molds are given. A 
collapsible center or core for bridges, culverts and sewers is fully illustrated with de 
tailed instructions for building. (No. 8 of Series.). 75 cents 

CONSTRUCTING CONCRETE PORCHES. By A. A. Houghton. 

A number of designs with working drawings of molds are fully explained so anyone 
can easily construct different styles of ornamental concrete porches without the pur¬ 
chase of expensive molds. (No. 9 of Series.).75 cents 

MOLDING CONCRETE FLOWER-POTS, BOXES, JARDINIERES, ETC. 

By A. A. Houghton. 

The molds for producing many original designs of flower-pots, urns, flower-boxes, 
jardinieres, etc., are fully illustrated and explained, so the worker can easily construct 
and operate same. (No. 10 of Series.). 75 cents 

MOLDING CONCRETE FOUNTAINS AND LAWN ORNAMENTS. By A. 

A. Houghton. 

The molding of a number of designs of lawn seats, curbing, hitching posts, pergolas, sun 
dials and other forms of ornamental concrete for the ornamentation of lawns and gar¬ 
dens, is fully illustrated and described. (No. 11 of Series;.75 cents 

CONCRETE FROM SAND MOLDS. By A. A. Houghton. 

A Practical Work treating on a process which has heretofore been h^id as a trade secret 
by the few who possessed it, and which will successfully mold every and any class of 
ornamental concrete work. The process of molding concrete with sand molds is of 
the utmost practical value, possessing the manifold advantages of a low cost of molds, 
the ease and rapidity of operation, perfect details to all ornamental designs, density 
and increased strength of the concrete, perfect curing of the work without attention 
and the easy removal of the molds regardless of any undercutting the design may have. 
192 pages. Fully illustrated. Price.$S.OO 

ORNAMENTAL CONCRETE WITHOUT MOLDS. By A. A. Houghton. 

The process for making ornamental concrete without molds has long been held as a 
secret, and now, for the first time, this process is given to the public. The book 
reveals the secret and is the only book published which explains a simple, practical 
method whereby the concrete w'orker is enabled, by employing w’ood and metal tem¬ 
plates of different designs, to mold or model in concrete any Cornice, Archivolt, 
Column, Pedestal, Base Cap, Urn or Pier in a monolithic form—right upon the job. 
These may be molded in units or blocks, and then built up to suit the specifications 
demanded. This work is fully illustrated, with detailed engravings. Price . $2.00 

CONCRETE FOR THE FARM AND IN THE SHOP. By H. Colii 
Campbell, C.E., E.M. 

A new book illustrating and describing in plain, simple language many of the 
numerous applications of concrete within the range of the homo worker. Among the 
subjects treated are: 

Principles of reinforcing; methods of protecting concrete so as to insure proper harden¬ 
ing: home-made mixers: mixing by hand and macliine: form construction, described 
and illustrated by drawings and photographs; construction of concrete w^alls and 
fences: concrete fence posts; concrete gate posts; corner posts; clothes line posts; 
grape arbor posts; tanks; troughs; cisterns; hog wallows; feeding floors and barn¬ 
yard pavements; foundations; w^ell curbs and platforms; indoor floors: sidewalks; steps; 
concrete hotbeds and cold frames; concrete slab roofs; walls for buildings; repairing 
leaks in tanks and cisterns, etc., etc. A number of convenient and practical tables 
for estimating quantities, and some practical examples, are also given. (5 x 7). 
149 pages, 51 illustrations. Price.$1.00 


II 











CATALOGUE OF GOOD, PRACTICAL BOOKS 


POPULAR HANDBOOK FOR CEMENT AND CONCRETE USERS. By 

Myron H. Lewis. 

This is a concise treatise of the principles and methods employed in the manufacture 
and use of cement in all classes of modem works. The author has brought together 
in this work all the salient matter of interest to the user of concrete and its many 
diversified products. The matter is presented in logical and systematic order, clearly 
written, fully illustrated and free from involved mathematics. Everything of value to 
the concrete user is given, including kinds of cement employed in construction, concrete 
architecture, inspection and testing, waterproofing, coloring and painting, rules, tables, 
working and cost data. The book comprises thirty-three chapters, 430 pages, 126 
illustrations. Price. .$3.00 

WATERPROOFING CONCRETE. By Myron H. Lewis. 

Modern Methods of Waterproofing Concrete and Other Structures. A condensed 
statement of the Principles, Rules, and Precautions to be Observed in Waterproofing 
and Dampprooflng Structures and Structural Materials. Price .... 75 cents 

DICTIONARIES 

STANDARD ELECTRICAL DICTIONARY. By T. O’Conor Sloane. 

An indispensable work to all interested in electrical science. Suitable alike for the 
student and professional. A practical handbook of reference containing definitions in 
about 5000 distinct words, terms and phrases. The definitions are terse and concise 
and include every term used in electrical science. Recently issued. An entirely new 
edition. Should be in the possession of all who desire to keep abreast with the progress 
of this branch of science. Complete, concise and convenient. Nearly 800 pages. Nearly 
600 illustrations. New Revised and Enlarged Edition. Price.$5.00 

DIES—METAL WORK 


DIES: THEIR CONSTRUCTION AND USE FOR THE MODERN WORKING; 
OF SHEET METALS. By J. V.r Woodworth. 

A most useful book, and one which should be in the hands of all engaged in the pres- 
working of metals; treating on the Designing, Constmcting, and Use of Tools, Fixtures 
and Devices, together with the manner in which they should be used in the Power 
Press, for the cheap and rapid production of the great variety of sheet-metal articles j 
now in use. It is designed as a guide to the production of sheet-metal parts at thej 
minimum of cost with the maximum of output. The hardening and tempering of] 
Press tools and the classes of work which may be produced to the best advantage by 
the use of dies in the power press are fully treated. Its 505 illustrations show dies,, 
press fixtures and sheet-metal working devices, the descriptions of which are so clear andj 
practical that all metal-working mechanics will be able to understand how to design, 
construct and use them. Many of the dies and press fixtures treated were either, 
constructed by the author or under his supervision. Others were built by skilful] 
mechanics and are in use in large sheet-metal establishments and machine shops." 
6th Edition. 400 pages, 523 illustrations. Price.$3.50’ 

PUNCHES, DIES AND TOOLS FOR MANUFACTURING IN PRESSES. By 

J. V. Woodworth. 

This work is a companion volume to the author’s elementary work entitled “Dies, Theirs 
Construction and Use.” It does not go into the details of die-making to the extent of] 
the author’s previous book, but gives a comprehensive review of the field of operations* 
carried on by presses. A large part of the information given has been drawn from thej 
author’s personal experience. It might well be termed an Encvclopedia of Die-Making, j 
Punch-Making, Die-Sinking, Sheet-Metal Working, and Making of Special Tools, Sub-! 
presses. Devices and Alechanical Combinations for Punching, Cutting, Bending, Form-*^ 
ing. Piercing, Drawing, Compressing and Assembling Sheet-Metal Parts, and also Arti¬ 
cles of other Materials in Machine Tools. 3rd Edition. 483 pages, 702 illustrations.] 
Price. $4.5r 

I? 















CATALOGUE OF GOuD, PRACTICiX BOOKS 


DRAWING—SKETCHING PAPER 


PRACTICAL PERSPECTIVE. By Richards and Colvin. 

Shows just how to make all kinds of mechanical drawings in the only practical per¬ 
spective isometric. Makes everything plain so that any mechanic can understand 
a sketch or drawing in this way. Saves time in the drawing room, and mistakes in the 
shops. Contains practical examples of various classes of work. 4th Edition. $1.00 

LINEAR PERSPECTIVE SELF-TAUGHT. By Herman T. C. Kraus. 

Tills w'ork gives the theory and practice of Unear perspective, as used in architectural, 
engineering and mechanical drawings. Persons taking up the study of the subject 
by themselves will be able, by the use of the instruction given, to readily grasp the 
subject, and by reasonable practice become good perspective draftsmen. The arrange¬ 
ment of the book is good; the plate is on the left-hand, while the descriptive text 
follows on the opposite page, so as to be readily referred to. The drawings are on 
sufficiently large scale to show the work clearly and are plainly figured. There is 
included a self-explanatory chart which gives all information necessary for the thorough 
understanding of perspective. This chart alone is worth many times over the price of 
the book. 2d Revised and enlarged Edition.$3.00 

SELF-TAUGHT MECHANICAL DRAWING AND ELEMENTARY MACHINE 
DESIGN. By F. L. Sylvester, M.E., Draftsman, with additions by Erik 
Oberg, associate editor of “Machinery.” 

This is a practical treatise on Mechanical Drawing and Machine Design, comprising 
the first principles of geometric and mechanical drawing, workshop mathematics, 
mechanics, strength of materials and the calculations and design of machine details. 
The author’s aim has been to adapt this treatise to the requirements of the practical 
mechanic and young draftsman and to present the matter in as clear and concise a 
manner as possible. To meet the demands of this class of students, practically all the 
important elements of machine design have been dealt with, and in addition algebraic 
formulas have been explained, and the elements of trigonornetry treated in the manner 
best suited to the needs of the practical man. The book is divided into 20 chapters, 
and in arranging the material, mechanical drawing, pure and simple, has been taken 
up first, as a thorough understanding of the principles of representing objects facilitates 
the further study of mechanical subjects. This is followed by the mathematics neces¬ 
sary for the solution of the problems in machine design which are presented later, and 
a practical introduction to theoretical mechanics and the strength of materials. The 
various elements entering into machine design, such as cams, gears, sprocket-wheels. ‘ 
cone pulleys, bolts, screws, couplings, clutches, shafting and fiy-wheels, have been 
treated in such a way as to make possible the use of the work as a text-book for a 
continuous course of study. 333 pages, 218 engravings. Price. . . . $2.50 

A NEW SKETCHING PAPER. 

A new specially ruled paper to enable you to make sketches or drawings in isometric 
perspective without any figuring or fussing. It is being used for shop details as well 
as for assembly drawings, as it makes one sketch do the work of three, and no workman 
can help seeing just what is wanted. Pads of 40 sheets, 6x9 inches, 40 cents. Pads 
of 40 sheets, 9x12 inches, 75 cents; 40 sheets, 12x18, Price.$1.50 


ELECTRICITY 


ARITHMETIC OF ELECTRICITY. By Prof. T. O’Conor Sloane. 

A practical treatise on electrical calculations of all kinds reduced to a series of rules, all 
of the simplest forms, and involving only ordinary arithmetic: each rule illustrated 
bv one or more practical problems, with detailed solution of each one. This book is 
classed among the most useful works published on the science of electricity, covering 
as it does the mathematics of electricity in a manner that will attract the attention 
of those who are not familiar with algebraical formulas. 22nd Edition. 196 pages. 
Price.. 


13 











CATALOGUE OF GOOD, PRACTICAL BOOKS 


COMMUTATOR CONSTRUCTION. By Wm. Baxter, Jr. 

The business end of any dynamo or motor of the direct current type is the commutator. 
This book goes into the designing, building, and maintenance of commutators, shows 
how to locate troubles and how to remedy them; everyone who fusses with dynamos 
needs this. 5th Edition.35 cents 

CONSTRUCTION OF A TRANSATLANTIC WIRELESS RECEIVING SET. 

By L. G. Pacent and T. S. Curtis. 

A work for the Radio student who desires to construct and operate apparatus that 
will permit of the reception of messages from the large stations in Ei^ope with an 
aerial of amateur proportions. 36 pages. 23 illustrations, cloth. Price . 35 cents 

DYNAMO BUILDING FOR AMATEURS, OR HOW TO CONSTRUCT A 
FIFTY-WATT DYNAMO. By Arthur J. Weed, Member of N. Y. Electrical 
Society. 

A practical treatise showing in detail the construction of a small dynamo or motor, the 
entire machine work of which can be done on a small foot lathe. Dimensioned working 
drawings are given for each piece of machine work, and each operation is clearly 
described. This machine, when used as a dynamo, has an output of fifty w^atts ; when 
used as a motor it wdll drive a small drill press or lathe. It can be used to drive a 
sewing machine on any and all ordinary work. The book is illustrated with more 
than sixty original engravings showing the actual construction of the different parts. ]] 
Among the contents are chapters on; 1. Fifty-Watt Dynamo. 2. Side Bearing ■■ 
Rods. 3. Field Punching. 4. Bearings. 5. Commutator. 6. Pulley. 7. Brush 
Holders. 8. Connection Board. 9. Armature Shaft. 10. Armature. 11. Armature 
Winding. 12. Field Winding. 13. Connecting and Starting. Price, cloth, $1.00 


DESIGN DATA FOR RADIO TRANSMITTERS AND RECEIVERS. By 

Milton B. Sleeper. 

Far from being a collection of formulas, Design Data takes up in proper sequence the 
problems encountered in planning all types of receiving sets for short, medium and 
long wave work, and spark coil, transformer and vacuum tube transmitters operating 
on 200 meters. Tables have been worked out so that values can be found without 
the use of mathematics. Radio experimenters will find here information which will 
enable them to have the most modern and efficient equipment. Price . . 75 cents 


DYNAMOS AND ELECTRIC MOTORS AND ALL ABOUT THEM. By 

Edward Trevert. 

This volume gives practical directions for building a two H. P. Dvnamo of the Edison 
type capable of lighting about fifty mazda lamps of the 20-watt size. In addition, it 
gives directions for building two small electric motors suitable for running sewing 
machines. The concluding chapter describes the construction of a simple bichroma le 
battery adapted for running electric motors. 96 pages. Fully illustrated with detail 
drawings. Cloth. Price.$1.00 


ELECTRIC BELLS. By M. B. Sleeper. 

A complete treatise for the practical worker in installing, operating, and testing 
bell circuits, burglar alarms, thermostats, and other apparatus used with electric 
bells. Both the electrician and the experimenter will find in this book new material 
which is essential in their work. Tools, bells, batteries, unusual circuits, burglar 
alarms, annunciators, systems, thermostats, circuit breakers, time alarms, and other i 
apparatus used in bell circuits are described from the standpoints of their appUca- 
tion, construction, and repair. The detailed instructions for building the apparatus 
will appeal to the experimenter particularly. The practical worker will find the : 
chapters on Wiring Calculation of Wire Sizes and Magnet Windings, Upkeep of 
Systems and the Location of Faults of the greatest value in their work. 124 pages. 
Fully illustrated. Price.. cents i 

14 


I 









CATALOGUE OF GOOD, PRACTICAL BOOKS 


EXPERIMENTAL HIGH FREQUENCY APPARATUS — HOW TO MAKE 

AND USE IT. By Thomas Stanley Curtis. 

This book tells you how to build simple high frequency coils for experimental purpose 
in the home, school laboratory, or on the small lecture platform. The book is really 
a supplement to the same author’s “High Frequency Apparatus.” The experimental 
side only is covered in this volume, which is intended for those who want to build 
small coils giving up to an eighteen-inch spark. The book contains valuable in¬ 
formation for the physics or the manual training teacher who is on the lookout for 
interesting projects for his boys to build or experiment with. The apparatus is 
simple, cheap and perfectly safe, and with it some truly startling experiments may be 
performed. Among the contents are: Induction Coil Outfits Operated on Battery 
Current. Kicking Coil Apparatus. One-Half Kilowatt Transformer Outfit. Parts 
and Materials, etc., etc. 69 pages. Illustrated. Price.60 cents 

HIGH FREQUENCY APPARATUS, ITS CONSTRUCTION AND PRACTICAL 
APPLICATION. By Thomas Stanley Curtis. 

The most comprehensive and thorough work on this interesting subject ever produced. 
The book is essentially practical in its treatment and it constitutes an accurate record 
of the researches of its author over a period of several years, during which time dozens 
of coils were built and experimented with. The work has been divided into six basic 
parts. The first two chapters tell the uninitiated reader what the high frequency 
current is, what it is used for, and how it is produced. The second section, comprising 
four chapters, describes in detail the principles of the transformer, condenser, spark 
gap, and oscillation, transformer, and covers the main points in the design and con¬ 
struction of these devices as applied to the work in hand. The tliird section covers 
the construction of small high frequency outfits designed for experimental work in the 
home laboratory or in the classroom. The fourth section is devoted to electro- 
therapeutic and X-Ray apparatus. The fifth describes apparatus for the cultivatioc 
of plants and vegetables. The sixth section is devoted to a comprehensive discussion 
of apparatus of large size for use upon the stage in spectacular productions. The 
closing chapter, giving the current prices of the arts and materials required for the 
construction of the apparatus described, is included with a view to expediting the 
purchase of the necessary goods. The Second Edition includes much new matter 
along the line of home-made therapeutic outfits for physicians’ use. The matter on 
electro plant culture has also been elaborated upon. Second Revised and Enlarged 
Edition. 266 pages. New second edition. Fully illustrated. Price . $3.00 


ELECTRIC WIRING, DIAGRAMS AND SWITCHBOARDS. By Newton 
Harrison. 

A thoroughly practical treatise covering the subject of Electric Wiring in all its branches, 
including explanations and diagrams which are thoroughly explicit and greatly simplify 
the subject. Practical, every-day problems in wiring are presented and the method 
of obtaining inteUigent results clearly shown. Only arithmetic is used. Ohm’s law 
is given a simple explanation with reference to wiring for direct and alternatin'- 
currents. The fundamental principle of drop of potential in circuits is shown with its 
various apphcations. The simple circuit is developed with the position of mains, 
feeders and branches; their treatment as a part of a wiring plan and their employ 
ment in house wiring clearly illustrated. Some simple facts about testing are include*! 
in connection with the wiring. Molding and conduit work are given careful considera¬ 
tion; and switchboards are systematically treated, built up and illustrated, showing- 
the piupose they serve, for coniifjction with the circuits, and to shunt and corapoun*. 
wound machines. The simple principles of switchboard construction, the develop¬ 
ment of the sAvitchboard, the connections of the various instruments, including tho 
lightning arrester, are also plainly set forth. 

Alternating current wiring is treated, with explanations of the poAver factor, conditiors 
calling for various sizes of Avire, and a simple Avay of obtaining the sizes for single-phase, 
two-phase and three-phase circuits. This is the only complete work issued showing 
and telling you what you should know about direct and alternating current wiring. It 
is a ready reference The work is free froni advanced technicalities and mathematics, 
arithmetic being used throughout. It is in every respect a handy, well-written, 
instructive, comprehensive volume on wiring for the wireman, foreman, contractor, 
or electrician. 3rd edition, revised and enlarged. 315 pages; 137 illustraHon^ 
Price.. 





CATALOGUE OF GOOD, PRACTICAL BOOKS 


ELECTRIC TOY MAKING, DYNAMO BUILDING, AND ELECTRIC MOTOR 
CONSTRUCTION. By Prof. T. O’Conor Sloane. 

This work treats of the making at home of electrical toys, electrical apparatus, motors, 
dynamos and instruments in general, and is designed to bring within the reach of 
young and old the manufacture of genuine and useful electrical appliances. The work 
is especially designed for amateurs and young folks. 

Thousands of our young people are daily experimenting, and busily engaged in making 
electrical toys and apparatus of various kinds. The present work is just what is want¬ 
ed to give the much needed information in a plain, practical manner, with illustrations 
to make easy the carrying out of the work. 20th Edition. 210 pages, 77 illustrations. 
Price. $ 1.50 


ELECTRICIANS’ HANDY BOOK. By Prof. T. O’Conor Sloane. 

This work is intended for the practical electrician who has to make things go. The 
entire field of electricity is covered within its pages. Among some ot the subjects treated 
are: The Theory of the Electric Current and Circuit, Electro-Chemistry, Primary 
Batteries, Storage Batteries, Generation and Utilization of Electric Powers, Alter¬ 
nating Current, Arm.ature Winding, Dynamos and Motors, Motor Generators, 
Operation of the Central Station Switchboards, Safety Appliances, Distribution 
of Electric Light and Power, Street Mains, Transformers, Arc and Incandescent 
Lighting, Electric Measurements, Photometry, Electric Railways, Telephony, Bell- 
Wiring, Electric-Plating, Electric Heating, Wireless Telegraphy, etc. It contains no 
useless theory; everything is to the point. It teaches you just what you want to 
know about electricity. It is the standard work published on the subject. Forty- 
six chapters, 600 engravings. New 5th Edition, Revised and Enlarged. Price $ 4.00 


ELECTRICITY SIMPLIFIED. By Prof. T. O’Conor Sloane. 

The object of “Electricity Simplified” is to make the subject as plain as possible and 
to show what the modern conception of electricity is; to show how two plates of i 
different metal, immersed in acid, can send a message around the globe; to explain! | 
how a bundle of copper wire rotated by a steam engine can be the agent in lighting ! 
our streets, to teU what the volt, ohm and ampere are, and what high and low tensioB ' 
mean; and to answer the questions that perpetually arise in the mind in this age of ] 
electricity. 15th Revised Edition, 218 pages. Illustrated. Price . , $ 1.50 


EXPERIMENTAL WIRELESS STATIONS. By P. E. Edelman. 

The theory, design, construction and operation is fully treated including Wirelesn 
Telephony, Vacuum Tube, and quenched spark systems. The new enlarged edition j 
is just issued and is strictly up to date, correct and complete. This book tells - 
how to make apparatus to not only hear all telephoned and telegraphed radio mes¬ 
sages, but also how to make simple equipment that works for transmission over rea¬ 
sonably long distances. I'hen there is a host of new information included. The 
first and only book to give you all the recent important radio improvements, some i 
of which have never before been published. This volume anticipates every need of ! 
the reader who wants the gist of the art, its principles, simplifled calculations, appara¬ 
tus dimensions, and understandable directions for efficient operation. ; 

Vacuum tube circuits; amplifiers; long-distance sets; loop, coil, and underground : 
receivers; tables of wave-lengths, capacity, inductance; such are a few of the sub- ' 
jects presented in detail that satisfies. It is independent and one of the few that i 
describe all modern systems. i 

Endorsed by foremost instructors for its clear acciiracy, preferred by leading amateurs i 
for its dependable designs. The new experimental Wireless Stations is sure ! 
to be most satisfactory for your purposes. 27 chapters, 392 pages, 167 illustra¬ 
tions. Price . $ 3.00 ' 


RADIO TIME SIGNAL RECEIVER. By Austin C. Lescarboura. 

This new book, “A Radio Time Signal Receiver,” tells vou how to build a simple 1 
outfit designed expressly for the beginner. You can build the outfits in your own ‘ 
workshop and install them for jewelers either on a one-payment or a rental basis 
The apparatus is of such simple design that it may be made by the average amateur 5 
mechanic possessing a few ordinary tools. 42 pages. Paper. Price 35 cents j 

16 ' 1 






CATALOGUE OF GOOD, PRACTICAL BOOKS 


HOUSE WIRING. By Thomas W. Poppe. 

This work describes and illustrates the actual installation of Electric Light Wiring, 
the manner in which the work should be done, and the method of doing it. The book 
can be conveniently carried in the pocket. It is intended for the Electrician, Helps, 
and Apprentice. It solves all Wiring Problems and contains nothing that conflicts 
with the rulings of the National Board of Fire Underwriters. It gives just the informa¬ 
tion essential to the Successful Wiring of a Building. Among the subjects treated are 
Locating the Meter. Panel Boards. Switches. Plug Receptacles. Brackets. Ceiling 
Fixtures. The Meter Connections. The Feed Wires. The Steel Armored Cable 
System. The Flexible Steel Conduit System. The Ridig Conduit System. A digest 
of the National Board of Fire Underwriters’ rules relating to metallic wiring systems. 
Various switching arrangements explained and diagrammed. The easiest method or 
testing the Three- and Four-way circuits explained. The grounding of all metallic 
wiring systems and the reason for doing so shown and explained. The insulation of 
the metal parts of lamp fixtures and the reason for the same described and illustrated. 
208 pages. 4th Edition, revised and enlarged. 160 illustrations. Flexible cloth. 
Price..$1.00 

HOW TO BECOME A SUCCESSFUL ELECTRICIAN. By Prof. T. O’Conor 

Sloane. 

Every young man who wishes to become a successful electrician should read this book. 
It tells in simple language the surest and easiest way to become a successful electrician. 
The studies to be followed, methods of work, field of operation and the requirement, 
of the successful electrician are pointed out and fully explained. Every young en¬ 
gineer will find this an excellent stepping stone to more advanced works on electricity 
which he must master before success can be attained. Many young men become dis¬ 
couraged at the VRry outstart by attempting to read and study books that are far 
beyond their comprehension. This book serves as the connecting link between the 
rudiments taught in the public schools and the real study of electricity. It is inter¬ 
esting from cover to cover. 19th Revised Edition, just issued. 205 pages. Illus¬ 
trated. Price.$1.50 


RADIO HOOK-UPS. By Milton B. Sleeper. 

In this book the best circuits for different instruments and various purposes have been 
carefully selected and grouped together. All the best circuits for damped and un¬ 
damped wave receiving sets, buzzer,*spark coil and transformer sending equipment, as 
well as vacuum tube telegraph and telephone transmitters, wavemeters, vacuum tube 
measuring instruments, audibility meters, etc., are shown in this book. . 75 cents 


STANDARD ELECTRICAL DICTIONARY. By T. O’Conor Sloane. 

An indispensable work to all interested in electrical science. Suitable alike for the 
student and professional. A practical handbook of reference containing definitions 
of about 5,000 distinct words, terms and phra.ses. The definitions are terse and 
concise and include every term used in electrical science. Recently issued. An en¬ 
tirely new edition. Should be in the possession of all who desire to keep abreast with 
the progress of this branch of science. In its arrangement and typography the book 
is very convenient. The word or term defined is printed in black-faced type w'hich 
readily catches the eye, while the body of the page is in smaller but distinct type. The 
definitions are weU worded, and so as to be understood by the non-technical reader. 
The general plan seems to be to give an exact, concise definition, and then amplify 
and explain in a more popular way. Synonyms are also given, and references to other 
words and phrases are made. A very complete and accurate index of fifty pages is 
at the end of the volume; and as this index contains all synonyms, and as all phrases 
are indexed in every reasonable combination of words, reference to the proper place 
in the body of the book is readily made. It is difficult to decide how far a book of 
this character is to keep the dictionary form, and to what extent it may assume the 
encyclopedia form. For some purposes, concise, exactly worded definitions are needed; 
for other purposes, more extended descriptions are required. This book seeks to satisfy 
both demands, and does it with considerable success. Complete, concise and con¬ 
venient. 800 pages. Nearly 500 illustrations. New Revised and Enlarged Edition. 
Price.. ... ........ $5.00 


17 







CATALOGUE oF GOOD, PRACTICAL BOOKS 


STORAGE BATTERIES SIMPLIFIED. By Victor W. Page, M.S.A.E. 

A complete treatise on storage battery operating principles, repairs and applications. 
The greatly increasing application of storage batteries in modern engineering and 
mechanical work has created a demand for a book that will consider this subject 
completely and exclusively. This is the most thorough and authoritative treatise 
ever published on tlus subject. It is written in easily understandable, non-technical 
language so that anyone may grasp the basic principles of storage battery action as 
well as their practical industrial applications. All electric and gasohne automobiles 
use storage batteries. Every automobile repairman, dealer or salesman should have a 
good knowledge of maintenance and repair of these important elements of the motor 
car mechanism. Tliis book not only tells how to charge, care for and rebuild storage 
batteries but also outlines all the industrial uses. Learn how they rim street cars, 
locomotives and factory trucks. Get an understanding of the important functions they 
perform in submarine boats, isolated lighting plants, railway switch and signal systems, 
marine applications, etc. This book tells how they are used in central station standby 
service, for starting automobile motors and in ignition systems. Every practical use 
of the modern storage battery is outhned in this treatise 208 pages. P'ully illus- 
tirated. Price.$2.00 


TELEPHONE CONSTRUCTION, INSTALLATION, WIRING, OPERATION! 
AND MAINTENANCE. By W. H. Rad.cliffe and H. C. Cushing. _ J' 

This book is intended for the amateur, the wireman, or the engineer who desires to'!^ 
establish a means of telephonic communication between the rooms of his home, offlce.ii; 
or shop. It deals only with such things as may be of use to him rather than 
theories. 

Gives the principles of construction and operation of both the Bell and Independem 
instruments; approved methods of installing and wiring them; the means of protecting 
them from lightning and abnormal currents; their connection together for operation 
as series or bridging stations; and rules for their inspection and maintenance. Line 
wiring and the wiring and operation of special telephone systems are also treated. 

Intricate mathematics are avoided, and all apparatus, circuits and systems are thor¬ 
oughly described. The appendix contains definitions of units and terms used in the 
text. Selected wiring tables, which are very helpful, are also included. Among the 
subjects treated are Construction, Operation, and Installation of Telephone Instru¬ 
ments; Inspection and Maintenance of Telephone Instruments; Telephone Line 
Wiring; Testing Telephone Line Wires and Cables; Wiring and Operation of Special 
Telephone Systems, etc. 2nd Edition, revised and enlarged. 223 pages. 154 
illustrations...$1.50 


WIRELESS TELEGRAPHY AND TELEPHONY SIMPLY EXPLAINED. By 

Alfred P. Morgan. 

This is undoubtedly one of the most complete and comprehensible treatises on the . 
subject ever published, and a close study of its pages will enable one to master all the 
details of the wireless transmission of messages. The author has filled a long-felt 
want and has succeeded in furnishing a lucid, comprehensible explanation in simple 
language of the theory and practice of wireless telegraphy and telephony. 

Among the contents are: Introductory; Wireless Transmission and Reception—The 
Aerial System, Earth Connections—The Transmitting Apparatus, Spark Coils and 
Transformers. Condensers, Helixes, Spark Gaps, Anchor Gaps, Aerial Switches—The 
Receiving Apparatus, Detectors, etc.—Tuning and Coupling, Tuning Coils, Loose i 
Couplers, Variable Condensers, Directive Wave Systems—Miscellaneous Apparatus, 
Telephone Receivers, Range of Stations, Static Interference—Wireless Telephones, 
Sound and Sound Waves,The Vocal Cords and Ear—Wireless Telephone, How Sounds 
Are Changed into Electric Waves—Wireless Telephones, The Apparatus—Summary.. 
154 pages. 156 engravings. Price.$1.50 


WIRING A HOUSE. By Herbert Pratt. 

Shows a house already built; tells just how to start about wiring it; where to begin; 
what wire to use; how to run it according to Insin-ance Rules; in fact, just the informa¬ 
tion you need. Directions apply equally to a shop. Fourth edition . . 35 cents 

i8 







CATALOGUE OF GOOD, PRACTICAL BOOKS 


ELECTROPLATING 


IHE MODERN ELECTROPLATER. By Kenneth M. Coggeshall. 

This is one of the most complete and practical books on electroplating and allied 
processes that has been published as a text for the student or professional plater. 
It IS written in simple language and explains all details of electroplating in a concise 
yet complete manner. It starts at the beginning and gives an elementary outline 
of electricity and chemistry as relates to plating, then considers shop layout and 
equipment and gives all the necessary information to do reliable and profitable electro¬ 
plating in a modern commercial manner. Full instructions are given for the prepara¬ 
tion and finishing of the work and formulae and complete directions are included for 
making all kinds of plating solutions, many of these having been trade secrets until 
pubhshed in tliis instruction manual. Anyone interested in practical plating and 
metal finishing will find this book a valuable guide and complete manual of the art. 
Cloth. 142 illustrations. 270 pages. Price.$3.00 


FACTORY MANAGEMENT, ETC. 


MODERN MACHINE SHOP CONSTRUCTION, EQUIPMENT AND 
MANAGEMENT. By O. E. Perrigo, M.E. 

The only work published that describes the modern machine shop or manufacturing 
plant from the time the grass is growing on the site intended for it until the finished 
product is shipped. By a careful study of its thirty-six chapters the practical man 
may economically build, etflciently equip, and successfully manage the modern machine 
shop or manufacturing establishment. .Just the book needed i\v those contemplating 
the erection of modern shop buildings, the rebuilding and reorganization of old ones, 
or the introduction of modern shop methods, time and cost systems. It is a book 
written and illustrated by a practical shop man for practical shop men who are too 
busy to read theories and want facts. It is the most complete all-around book of its 
kind ever published. It is a practical book for practical men, from the apprentice in 
the shop to the president in the office. It minutely describes and illustrates the most 
simple and yet the most eflficient time and cost system yet devised. 384 pages. 219 
illustrations. Price.$5.00 


FUEL 


COMBUSTION OF COAL AND THE PREVENTION OF SMOKE. By Wm. 

M. Barr. 

This book has been prepared with special reference to the generation of heat by the 
combustion of the common fuels found in the United States, and deals particularly 
with the conditions necessary to the economic and smokeless combustion of bituminous 
coals in Stationary and Locomotive Steam Boilers. 

The presentation of this important subject is systematic and progressive. The ar¬ 
rangement of the book is in a series of practical questions to which are appended 
accurate answers, which describe in language, free from technicalities, the several 
processes involved in the furnace combustion of American fuels; it clearly states the 
essential requisites for perfect combustion, and points out the best methods for furnace 
construction for obtaining the greatest quantity of heat from any given quality of 
coal. 5th Edition. Nearly 350 pages, fully illustrated. Price. . . . $1.50 

19 















CATALOGUE OF GOOD, PRACTICAL BOOKS 


GAS AND OIL ENGINES 


THE GASOLINE ENGINE ON THE FARM; ITS OPERATION, REPAIR 
AND USES. By Xeno W. Putnam. 

This is a practical treatise on the Gasoline and Kerosene Engine intended for the man 
who wants to know just how to manage his engine and how to apply it to all kinds oi 
farm work to the best advantage. 

This book abounds with hints and helps for the farm and suggestions for the home 
and housewife. There is so much of value in this book that it is impossible to ade¬ 
quately describe it in such small space. Suffice to say that it is the kind of a book 
every farmer will appreciate and every farm home ought to have. Includes selecting 
the most suitable engine for farm work, its most convenient and efficient installation, 
with chapters on troubles, their remedies, and how to avoid them. The care and 
management of the farm tractor in plowing, harrowing, harvesting and road grading 
are fully covered; also plain directions are given for handling the tractor on the road. 
Special attention is given to relieving farm life of its drudgery by applying power to 
the disagreeable small tasks which must otherwise be done by hand. Many home¬ 
made contrivances for cutting wood, supplying kitchen, garden, and barn with water, 
loading, hauling and unloading hay, delivering grain to the bins or the feed trough 
are included; also full directions for making the engine milk the cows, churn, wash, 
sweep the house and clean the windows, etc. Very fully illustrated with drawings of 
working parts and cuts showing Stationary, Portable and Tractor Engines doing all 
kinds of farm work. All money-making farms utilize power. Learn how to utilize 
power by reading the pages of this book. It is an aid to the result getter, invaluable 
to the up-to-date farmer, student, blacksmith, implement dealer and, in fact, all who 
can apply practical knowledge of stationary gasoline engines or gas tractors to advan¬ 
tage. 530 pages. Nearly 180 engravings. Price.$3.00 


GASOLINE ENGINES: THEIR OPERATION, USE AND CARE. By A. Hyatt 
Verrill. 

The simplest, latest and most comprehensive popular work published on Gasoline 
Engines, describing what the Gasoline Engine is; its construction and operation; how 
to install it; how to select it; how to use it and how to remedy troubles encountered. 
Intended for Owners, Operators and Users of Gasoline Motors of all kinds. This 
work fully describes and illustrates the various types of Gasoline Engines used in 
Motor Boats, Motor Vehicles and Stationary Work. The parts, accessories and 
appliances are described, with chapters on ignition, fuel, lubrication, operation and 
engine troubles. Special attention is given to the care, operation and repair of motors, 
with useful hints and suggestions on emergency repairs and makeshifts. A complete 
glossary of technical terms and an alphabetically arranged table of troubles and their 
symptoms form most valuable and unique features of this manual. Nearly every 
illustration in the book is original, having been made by the author. Every page is 
full of interest and value. A book which you cannot afford to be without, 275 pages. 
152 specially made engravings. Price.$2.00 


SAS, GASOLINE, AND OIL ENGINES. By Gardner D. Hiscox. 

Just issued, 23d revised and enlarged edition. Every user of a gas engine needs this 
book. Simple, instructive, and right up-to-date. Thp only complete work on the 
subject. Tells all about the running and management of gas, gasoline and oil engines, 
as designed and manufactured in the United States. Explosive motors for stationary 
marine and vehicle power are fully treated, together with illustrations of their parts 
and tabulated sizes, also their care and running are included. Electric ignition by 
induction coil and jump spark are fully explained and illustrated, including valuable 
information on the testing for economy and power and the erection of power plants. 

The rules and regulations of the Board of Fire Underwriters in regard to the installation 
and management of gasoline motors are given in full, suggesting the safe installation 
of explosive motor power, A list of United States Patents issued on gas, gasoline and 
oil engines and their adjuncts from 1875 to date is included. 640 pages. 435 engrav¬ 
ings. Folding plates. Price.$3.00 


20 











CATALOGUE OF GOOD, PRACTICAL BOOKS 


GAS ENGINES AND,PRODUCER-GAS PLANTS. By R. E. Mathot, M.E. 

This is a practical treatise, setting forth the principles of gas engine and producer 
design, the selection and installation of an engine, conditions of perfect operation, 
producer-gas engines and their possibilities: the care of gas engines and producer-gas 
plants, with a chapter on volatile hydrocarbon and oil engines. A practical guide for 
the gas engine designer, Liser and engineer in the construction, selection, purchase, in¬ 
stallation, operation and maintenance of gas engines. Every part of the gas engine is de¬ 
scribed in detail, tersely, clearly and with a thorough understanding of the requirements of 
the mechanic. Recognizing the need of a volume that would assist the gas engine 
user in understanding the motor upon which he depends for power, the author has 
discussed the subject without the help of any mathematics. Helpful suggestions as to 
the purchase of an engine, its installation, care and operation, form a most valuable 
feature of the book. 6x9inches. Cloth. 314 pages. 152 illustrations. Price.. $3.00 

GAS ENGINE CONSTRUCTION, OR HOW TO BUILD A HALF-HORSE¬ 
POWER GAS ENGINE. By Parsell and Weed. 

A practical treatise of 300 pages describing the theory and principles of the action of 
Gas Engines of various types and the design and construction of a half-horse-power 
Gas Engine, with illustrations of the work in actual progress, together with the dimen¬ 
sioned working drawings, giving clearly the sizes of the various details; for the student, 
the scientific investigator, and the amateur mechanic. This book treats of the subject 
more from the standpoint of practice than that of theory. The principles of operation 
of Gas Engines are clearly and simply described, and then the actual construction of a 
half-horse-power engine is taken up, step by step, showing in detail the making of the 
Gas Engine. 3d Edition. 300 pages. Price.$3.00 

HOW TO RUN AND INSTALL GASOLINE ENGINES. By C. Von Gulin. 

Revised and enlarged edition just issued. The object of this little book is to furnish 
a pocket instructor for the beginner, the busy man who uses an engine for pleasure or 
profit, but who does not have the time or inclination for a technical book, but simply 
to thoroughly understand how to properly operate, install and care for his own engine. 
The index refers to each trouble, remedy, and subject alphabetically. Being a quick 
reference to find the cause, remedy and prevention for troubles, and to become an 
expert with his own engine. Pocket size. Paper binding. Price . . 26 cents 


GEARING AND CAMS 


BEVEL GEAR TABLES. By D. Ag. Engstrom. 

A book that will at once commend itself to mechanics and draftsmen. Does away 
with all the trigonometry and fancy figuring on bevel gears, and makes it easy for any¬ 
one to lay them out or make them just right. There are 36 full-page tables that 
show every necessary dimension for all sizes or combinations you’re apt to need. No 
puzzling, figuring or guessing. Gives placing distance, all the angles (including 
cutting angles), and the correct cutter to use. A copy of this prepares you for any¬ 
thing in the bevel-gear line. 3d Edition. 66 pages.$1.60 

dlHANGE GEAR DEVICES. By Ogcar E. Perrigo. 

A practical book for every designer, draftsman, and mechanic interested in the inven¬ 
tion and development of the devices for feed changes on the different machines requir¬ 
ing such mechanism. All the necessary information on this subject is taken up. 
analyzed, classified, sifted, and concentrated for the use of busy men who have not the 
time' to go through the masses of irrelevant matter with which such a subject is usu¬ 
ally encumbered and select such information as will be useful to them. 

It shows just what has been done, how it has been done, when it was done, and who 
did it. It saves time in hunting up patent records and re-inventing old ideas. 3rd 
Edition. 101 pages.$1.60 

DRAFTING OF CAMS. By Louis Rouillion. 

The laying out of cams is a serious problem unless you know how to go at it right. 
This puts you on the right road for practically any kind of cam you are hkely to run 
up against. Sd Edition.86 cenli 


21 










CATALOGUE OF GOOD, PRACTICAL BOOKS 




HYDRAULICS 


HYDRAULIC ENGINEERING. By Gardner D. Hiscox. 

A treatise on the properties, power, and resources of water for all purposes. Including 
the measurement of streams, the flow of water in pipes or conduits; the horse-power 
of falling water, turbine and impact water-wheels, wave motors, centrifugal, recipro¬ 
cating and air-lift pumps. With .300 figures and diagrams and 36 practical tables. 
All who are interested in water-works devt'iopmenb will find this book a useful ono, 
because it is an entirely practical treatise upon a subject of present importance, and 
cannot fail in having a far-reaching influence, and for this reason should have a place 
in the working library of every engineer. Among the subjects treated are: Historical 
Hydraulics, Properties of Water, Measurement of the Flow of Streams; Flow¬ 
ing AVa ter Suface Orifices and Nozzles: Flow of AVater in Pipes; Siphons of Variom;/ 
Kinds; Dams and Great Storage Reservoirs: City and Town Water Supply; Wella 
and Their Reinforcement: Air Lift Methods of Raising Water; Artesian Wells, 
Irrigation of Ariel Districts; Water Power; Water-Wheels; Pumps and Pumping 
Machinery; Reciprocating Pumps; Hydraulic Power Transmission; Hydraulic 
Mining; Canals; Dredges: Conduits and Pipe Lines; Marine Hydraulics; Tidal and 
Sea Wave Power, etc. 320 pages. Price. . $4.60 


ICE AND REFRIGERATION 


POCKETBOOK OF REFRIGERATION AND ICE MAKING. By A. J. 

Wallis-Taylor. 

This is one of the latest and most comprehensive reference books published on the 
subject of refrigeration and cola storage. It explains the properties and refrigerating 
effect of the different fluids in use, the management of refrigerating machinery and the 
construction and insulation of cold rooms with their required pipe surface for different 
degrees of cold; freezing mixtures and non-freezing brines, temperatures of cold rooms 
for all kinds of provisions, cold storage charges for all classes of goods, ice making 
and storage of ice, data and memoranda for constant reference by refrigerating engineers, 
with nearly one hundred tables containing valuable references to every fact and con¬ 
dition required in tne installment and operation of a refrigerating plant. New 
edition just published. Price. $2.00 


INVENTIONS—PATENTS 


INVENTORS’ MANUAL, HOW TO MAKE A PATENT PAY. 

This is a book designed as a guide to inventors in perfecting their inventions, taking 
out their patents and disposing of them. It is not in any sense a Patent Solicitor’s 
Circular nor a Patent Broker’s Advertisement. No advertisements of any description 
appear in the work. It is a book containing a quarter of a century’s experience of a 
successful inventor, together with notes based upon the experience of many other 
inventors. 

Among the subjects treated in this work are; How to Invent. How to Secure a 
Good Patent. Value of Good Invention. How to Exhibit an Invention. How to 
Interest Capital. How to Estimate the Value of a Patent. Value of Design Patents. 
Value of Foreign Patents. Value of Small Inventions. Advice on Selling Patents. 
Advice on the Formation of Stock Companies. Advice on the Formation of Limited 
Liability Companies. Advice on Disposing of Old Patents. Advice as to Patent 
Attorneys. Advice as to Selling Agents. Forms of Assignments. License and Con¬ 
tracts. State Laws Concerning Patent Rights. 1900 Census of the United States by 
Counts of Over 10.000 Population. New revised and enlarged edition. 144 pages. 
Illustrated. Price. $1.60 


22 
















CATALOGUE OF GOOD, PRACTICAL BOOKS 


KNOTS 


KNOTS, SPLICES AND ROPE WORK. By A. Hyatt Verrill. 

This is a practical book giving complete and simple directions for making all the m08( 
useful and ornamental knots in common use, with chapters on Splicing, Pointing, 
Seizing, Serving, etc. This book is fully illustrated with one hundred and fifty 
original engravings, which show how each knot, tie or splice is formed, and its appear* 
ance when finished. The book will be found of the greatest value to Campers, Yachts¬ 
men, Travelers, Boy Scouts, in fact, to anyone having occasion to use or handle rope 
or knots for any purpose. The book is thoroughly reliable and practical, and is not 
only a guide, but a teacher. It is the standard work on the subject. Among the 
contents are: 1. Cordage, Kinds of Rope. Construction of Rope, Parts of Rope 
Cable and Bolt Rope. Strength of Rope, Weight of Rope. 2. Simple Knots anu 
Bends. Terms Used in Handling Rope. Seizing Rope. 3. Ties and Hitches. 4. 
Noose, Loops and Mooring Knots. 5. Shortenings, Grommets and Salvages. 0. 
Lashings, Seizings and Splices. 7. Fancy Knots and Rope Work. 104 pages. 154 
original engravings. Price. $1.00 


LATHE WORK 


LATHE DESIGN, CONSTRUCTION, AND OPERATION, WITH PRACTICAL 
EXAMPLES OF LATHE WORK. By Oscar E. Perrigo. 

A new revised edition, and the only complete American work on the subject, written 
by a man who knows not only how work ought to be done, but who also knows how 
to do it, and how to convey this knowledge to others. It is strictly up-to-date in its 
descriptions and illustrations. Lathe history and the relations of the lathe to manu¬ 
facturing are given; also a description of the various devices for feeds and thread 
cutting mechanisms from early efforts in this direction to the present time. Lathe 
design is thoroughly discussed, including back gearing, driving cones, thread-cutting 
gears, and all the essential elements of the modern lathe. TL he classification of lathes 
is taken up, giving the essential differences of the several types of lathes including, 
as is usually understood, engine lathes, bench lathes, speed lathes, forge lathes, gap 
lathes, pulley lathes, forming lathes, multiple-spindle lathes, rapid-reduction lathes, 
precision lathes, turret lathes, special lathes, electrically-driven lathes, etc. In addi¬ 
tion to the complete exposition on construction and design, much practical matter on 
lathe installation, care and operation has been incorporated in the enlarged 1915 edi¬ 
tion. All kinds of lathe attachments for drilling, milling, etc., are described and 
complete instructions are given to enable the novice machinist to grasp the art of lathe 
operation as well as the principles involved in design. A number of difficult machining 
operations are described at length and illustrated. The new edition has nearly 500 
pages and 350 illustrations. Price.. $3.00 

LATHE WORK FOR BEGINNERS. By Raymond Francis Yates. 

A simple, straightforward text-book for those desiring to learn the operation of a 
wood-turning or metal-turning lathe. The first chapter tells how' to choose a lathe 
and all of the standard types on the market are described. Simple and more advanced 
lathe work is thoroughly covered and the operation of all lathe attachments such as 
millers, grinders, polishers, etc., is described. The treatment starts from the very 
bottom and leads the reader through to a point where he will be able to handle the 
larger commercial machines with very little instruction. The last chapter of the 
book is devoted to things t-o make on the lathe and includes a model rapid-fire naval 
gun. This is the only book published in this country that treats lathe work from 
the standpoint of the amateur mechanic. 162 illustrations. About 250 pag'js, 12mo. 
Price. $2.00 

TURNING AND BORING TAPERS. By Fred H. Colvin. 

There are two ways to turn tapers; the right way and one other. This treatise has 
to do with the right way; it tells you how to start the work properly, how to set the 
lathe, what tools to use and how to use them, and forty and one other little things 
that you should know. Fift^ edition. Price .. 35 cents 

23 












CATALOGUE OE GOOD, PRACTICAL BOOKS 


LIQUID AIR 


LIQUID AIR AND THE LIQUEFACTION OF GASES. By T. O’Conor Sloane. 

This book gives the history of the theory, discovery, and manufacture of Liquid Air, 
and contains an illustrated description of all the experiments that have excited the 
wonder of audiences all over the country. It shows how liquid air, like water, is 
carried hundreds of miles and is handled in open buckets. It tells what may be ex¬ 
pected from it in the near future. 

\ book that renders simple one of the most perplexing chemical problems of the 
century. Startling developments illustrated by actual experiments. 

It is not only a wmrk of scientific interest and authority, but is intended for the general 
reader, being written in a popular style—easily understood by everyone. Third 
eaition. Revised and Enlarged. 394 pages. New Edition. Price . . . $3.00 


LOCOMOTIVE ENGINEERING 


AIR-BRAKE CATECHISM. By Robert H. Blackall. 

This book is a standard text-book. It covers the Westinghouse Air-Brake Equipment, 
including the No. 5 and the No. 6 E. T. Locomotive Brake Equiprrent; the K (Quick 
Service) Triple Valve for Freight Service: and the Cross-Compound Pump. The 
operation of all parts of the apparatus is explained in detail, and a practical way of 
finding their peculiarities and defects, with a proper remedy, is given. It contains 
2,000 questions with their answers, which will enable any railroad man to pass any 
examination on the subject of Air Brakes. Endorsed and used by air-brake instruc¬ 
tors and examiners on nearly every railroad in the United States. 28th Edition. 411 
pages, fully illustrated with colored plates and diagrams. Price.$2.50 

COMBUSTION OF COAL AND THE PREVENTION OF SMOKE. By Wm. 

M. Barr. 

This book has been prepared with special reference to the generation of heat by tha 
combustion of the common fuels found in the United States and deals particularly 
with the conditions necessary to the economic and smokeless combustion of bituminous 
coal in Stationary and Locomotive Steam Boilers. 

Presentation of this important subject is systematic and progressive. The ar¬ 
rangement of the book is in a series of practical questions to which are appended 
accurate answers, which describe in language free from technicalities the several 
processes involved in the furnace combustion of American fuels; it clearly states the 
essential requisites for perfect combustion, and points out the best methods of furnace 
construction for obtaining the greatest quantity of heat from any given quality of 
coal. Nearly 350 pages, fully illustrated. Price.$1.50 

DIARY OF A ROUND-HOUSE FOREMAN. By T. S. Reilly. 

This is the greatest book of railroad experiences ever published. Containing a fund of 
information and suggestions along the line of handling men, organizing, etc., that one 
cannot afford to miss. 158 pages. Price.$1.50 

LINK MOTIONS, VALVES AND VALVE SETTING. By Fred H. Colvin, 
Associate Editor of American Machinist. 

A handy book for the engineer or machinist that clears up the mysteries of valve 
setting. Shows the different valve gears in use. how they work, and why. Piston 
and slide valves of different types are illustrated and explained. A book that every 
railroad man in the motive power department ought to h^e. Contains chapters on 
Locomotive Link Motion, Valve Movements, Setting Slide Valves, Analysis by 
Diagrams, Modern Practice, Slip of Block, Shce Valves, Piston Valves, Setting Piston 
Valves, Joy-Alien Valve Gear, Walschaert Valve Gear, Gooch Valve Gear, Alfree- 
Hubbell Valve Gear, etc., etc. 3rd Edition, 101 »ages. Fully illustrated. Price 

75 cents 


24 















CATALOGUE OF GOOD, PRACTICAL BOOKS 


LOCOMOTIVE BOILER CONSTRUCTION. By Frank A. Klbinhans. 

The construction of boilers in general is treated, and, following this, the locomotive 
boiler is taken up in the order in which its various parts go through the shop. Shows 
all types of boilers used; gives details of construction; practical facts, such' as life of 
riveting, punches and dies; work done per day, allowance for bending and flanging 
sheets, and other data. Including the recent Locomotive Boiler Inspection Laws 
and Examination Questions with their answers for Government Inspectors. Contains 
chapters on Laying Out Work; Flanging and Forging; Punching; Shearing; Plate 
Planing; General Tables; Fini.shing Parts; Bending; Machinery Parts; Riveting; 
Boiler Details; Smoke Box Details; Assembling and Calldng; Boiler Shop 
Machinery, etc., etc. 

There isn’t a man who has anything to do with boiler work, either new or repair work, 
who doesn’t need this book. The manufacturer, superintendent, foreman, and boiler 
worker—all need it. No matter what the type of boiler, you’ll find a mint of informa¬ 
tion that you wouldn't be without. 451 pages, 334 illustrations, five large folding 
plates. Price.$3.50 

I 

LOCOMOTIVE BREAKDOWNS AND THEIR REMEDIES. By Geo. L. 

Fowler. Revised by Wm. W. Wood, Air-Brake Instructor. Just issued. 
Revised pocket edition. 

It is out of the question to try and tell you about every subject that is covered in this 
pocket edition of Locomotive Breakdowns. Just imagine all the common troubles 
that an engineer may expect to happen some time, and then add all of the unexpected 
ones, troubles that could occur, but that you have never thought about, and you will 
find that they are all treated with the very best methods of repair. Walschaert 
Locomotive Valve Gear Troubles, Electric Headlight Troubles, as well as Questions 
and Answers on the Air Brake are all included. 293 pages. 8th Revised Edition. 
Fully illustrated. $1.50 


LOCOMOTIVE CATECHISM. By Robert Grimshaw. 

The 30th revised and enlarged edition, just off the press, is a new boo’ic from cover 
to cover. It is bigger, better, more authoritative, and useful than ever. 11 is decidedly 
the best work on this subject ever published. It puts not only the underlying prin¬ 
ciples, but the practical handling and operation of all kinds of Locomotives at your 
finger tips. Answers over four thousand questions about Steam and Electric Loco¬ 
motives, and all kinds of Air Brakes. Specially helpful to all preparing for an exami¬ 
nation. You can get more valuable, up-to-date information from this book, and 
get it more quickly and easily, than from any other source; and the price is within 
reach of every engineer, fireman and apprentice. Contains four thousand examina¬ 
tion questions with their answers and is written in such simple language that all 
can imderstand it. An absolute authority on all subjects relating to the Locomotive. 
A self-educator on the Locomotive without an equal. It has been highly endorsed 
by the Brotherhood Journals and by thousands of practical Railroaders. It contains 
just the questions that will be asked of you when examined for promotion. It tells 
at once not only what to do but what not to do. 1000 pages. 468 illustrations 
Price.$4.00 

PREVENTION OF RAILROAD ACCIDENTS, OR SAFETY IN RAILROADING. 

By George Bradshaw. 

This book is a heart-to-heart tallc with Railroad Employees, dealing with facts, not 
theories, and showing the men in the ranks, from every-day experience, how accidents 
occur and how they may be avoided. The book is illustrated with seventy original 
photographs and drawings showing the safe and unsafe methods of work. No vision¬ 
ary schemes, no ideal pictures. Just plain facts and Practical Suggestions are given. 
Every railroad employee who reads the book is a better and safer man to have in 
railroad service. It gives just the information which Tvill be the means of preventing 
many injuries and deaths. All railroad employees should procure a copy; read it, 
and do your part in preventing accidents. 169 pages. Pocket size. Fully illustrated. 
Price.. . 50 cents 









CATALOGUE OF GOOD, PRACTICAL BOOKS 


I 


TRAIN RULE EXAMINATIONS MADE EASY. By G. E. CollingwoOd. 

This is the only practical-work on train rules in print. Every detail is covered, and 
puzzling points are explained in simple, comprehensive language, making it ? practical ' 
treatise for the Train Dispatcher, Engineman, Trainman, and all others who have to . 
do with the movements of trains. Contains complete and reliable information of the 
Standard Code of Train Rules for single track. Shows Signals in Colors, as used on 
the different roads. Explains fully the practical application of train orders, giving a | 
clear and definite understanding of all orders which may be used. The meaning and j 
necessity for certain rules are explained in such a manner that the student may know 
beyond a doubt the rights conferred under any orders he may receive or the action 
required by certain ruies. As nearly all roads require trainmen to pass regular exami ] 

nations, a complete set of examination questions, with their answers, are included. i 

These will enable the student to pass the required examinations with credit to himself I 
and the road for which he works. Second Edition revised. 234 pages. Fully illus- . 
trated with Train Signals in Colors. Price.$1.50 


THE WALSCHAERT AND OTHER MODERN RADIAL VALVE GEARS FOR 
LOCOMOTIVES. By Wm. W. Wood. 

If you would thoroughly understand the Walschaert Valve Gear you should possess a 
copy of this book, as the author takes the plainest form of a steam engine—a stationary 
engine in the rough, that will only turn its crank in one direction—and from it builds 
up—with the reader’s help—a modern locomotive equipped with the Walschaert 
Valve Gear, complete. The points discussed are clearly illustrated; two large folding 
plates that show the positions of the valves of both inside or outside admission type, as 
well as the links and other parts of the gear when the crank is at nine different points 
in its revolution, are especially valuable in making the movement clear. These employ 
sliding cardboard models which are contained in a pocket in the cover. 

The book is divided into five general divisions, as follows; 1. Analysis of the gear. 
2. Designing and erecting the gear. 3. Advantages of the gear. 4. Questions and 
answers relating to the Walschaert Valve Gear. 5. Setting valves with the Wal¬ 
schaert Valve Gear; the tlmee primary types of locomotive valve motion: modern 
radial valve gears other than the AValschaert; the Hobart All-free Valve and Valve 
Gear, with questions and answers on breakdowns; the Baker-Pilliod Valve Gear; the 
Improved Baker-Pilhod Valve Gear, with questions and answers on breakdowns. 

The questions with full answers given will be especially valuable to firemen and engi¬ 
neers in preparing for an examination for promotion. 245 pages. Fully illustrated. 
Third Revised New Edition. Price.$2.50 

WESTINGHOUSE E-T AIR-BRAKE INSTRUCTION POCKET BOOK. By 

Wm. W. Wood, Air-Brake Instructor. 

Here is a book for the railroad man, and the man who aims to be one. It i6 without 
doubt the only complete work published on the Westinghouse E-T Locomotive Brake 
Equipment. Written by an Air-Brake Instructor who knows just what is needed. It 
covers the subject thoroughly. Everything about the New Westinghouse Engine and 
render Brake Equipment, including the standard No. 5 and the Perfected No. 6 
style of brake, is treated in detail. Written in plain English and profusely illustrated 
with Colored Plates, which enable one to trace the flow of pressures throughout the 
entire equipment. The best book ever published on the Air Brake. Equally good for 
the beginner and the advanced engineer. Will pass anyone through any examination. 
It informs and enhghtens you on every point. Indispensable to every engineman and 
trainman. 

Contains examination questions and answers on the E-T equipment. Covering what 
the E-T Brake is. How it should be operated. What to do when defective. Not a 
question can bo asked of the engineman up for promotion, on either the No. 5 or the 
No. 6 E-T equipment, that is not asked and answered in the book. If you want to 
thoroughly understand the E-T equipment get <a copy of this book. It covers every 
detail. Makes Air-Brake troubles and examinations easy. Second Revised and 
Enlarged Edition. Price.•.$2.50 


26 









CATALOGUE OF GOOD, PRACTICAL BOOKS 


« 


MACHINE-SHOP PRACTICE 


AMERICAN TOOL P/IAKING AND INTERCHANGEABLE MANUFACTUR¬ 
ING. By J. V. Woodworth. 

A "shoppy” book, containing no theorizing, no problematical or experimental devices, 
there are no badly proportioned and impossible diagrams, no catalogue cuts, but a 
valuable collection of drawings and descriptions of devices, the rich fruits of the author’s 
own experience. In its 500-odd pages the one subject only. Tool Making, and what¬ 
ever relates thereto, is dealt with. The work stands without a rival. It is a complete 
practical treatise on the art of American Tool Making and system of interchangeable 
manufacturing as carried on to-day in the United States. In it are described and 
illustrated all of the different types and classes of small tools, fixtures, devices, and 
special appliances which are in general use in all machine-manufacturing and metal¬ 
working establishments where economy, capacity, and interchangeability in the p’'~ 
duction of machined metal parts are imperative. The science of jig making is exhaus¬ 
tively discussed, and particular attention is paid to drill jigs, boring, profiling and milling 
fixtures and other devices in which the parts to be machined are located and fastened 
within the contrivances. All of the tools, fixtures, and devices illustrated and de¬ 
scribed have been or are used for the actual production of work, such as parts of drill 
presses, lathes, patented machinery, typewriters, electrical apparatus, mechanical ap¬ 
pliances, brass goods, composition parts, mould products, sheet metal articles, drop- 
forgings, jewelry, watches, medals, coins, etc. 3rd Edition. 531 pages. Price $4.50 

MACHINE-SHOP ARITHMETIC. By Colvin-Cheney. 

This is an arithmetic of the things you have to do with daily. It tells you plainly 
about: how to find areas in figures; how to find surface or volume ol balls or spheres; 
handy ways for calculating; about compound gearing; cutting screw threads on any 
lathe; driUing for taps; speeds of drills; taps, emery wheels, grindstones, milling 
cutters, etc.; all about the Metric system with conversion tables; properties of metals; 
strength of bolts and nuts; decimal equivalent of an inch. All sorts of machine-shop 
figuring and 1,001 other things, any one of which ought to be worth more than 
the price of this book to you, and it saves you the trouble of bothering the boss. 7th 
edition. 131 pages. Price.75 cents 

MODERN MACHINE-SHOP CONSTRUCTION, EQUIPMENT AND MAN¬ 
AGEMENT. By Oscar E. Perrigo. 

The only work published that describes the Modern Shop or Manufacturing Plant 
from the time the grass is growing on the site intended for it until the finished product 
is shipped. Just the book needed by those contemplating the erection of modern shop 
buildings, the rebuilding and reorganization of old ones, or the introduction of Modern 
Shop Methods, time and cost systems. It is a book written and illustrated by a prac¬ 
tical shop man for practical shop men who are too busy to read theories and want facts. 
It is the most complete all-round book of its kind ever published.. 384 pages. 
219 original and specially-made illustrations. Revised and Enlarged Edition. 
Price ..$5.00 

** SHOP KINKS.” By Robert Grimshaw. 

A book of 400 pages and 222 illustrations, being entirely different from any other 
book on machine-shop practice. Departing^ from conventional style, the^ author 
avoids universal or common shop usage and limits his work to showing special ways 
of doing things better, more cheaply and more rapidly than usual. As a result the 
advanced methods of representative establishments of the world are placed at the 
disposal of the reader. This book shows the proprietor where large savings are possible, 
and how products may be improved. To the employee it holds out suggestions that, 
properly applied, will hasten his advancement. No shop can afford to be without it. 
It bristles with valuable wrinkles and helpful suggestions. It will benefit all, from 
apprentice to proprietor. 5th edition. Price.$3.00 

THREADS AND THREAD-CUTTING. By Colvin and Stabel. 

This clears up many of the mysteries of thread-cutting, such as double and triple 
threads, internal threads, catching threads, use of hobs, etc. Contains a lot of useful 
hints and several tables. 4th edition. Price .. cents 

27 










CATALOGUE OF GOOD, PRACTICAL BOOKS 


THE WHOLE FIELD OF MECHANICAL MOVEMENTS 
COVERED BY MR. HISCOX’S TWO BOOKS 


We publish two books by Gardner D. Hiscoz that will keep you from 'inventing things 
that have been done before, and suggest ways of doing things that you have not thought oj 
before. Many a man spends time and money, pondering over some mechanical problem, 
only to learn, after he has solved the problem, that the same thing has been accomplished 
and put in practice by others long before. Time and money spent in an effort to accom¬ 
plish what has already been accomplished are time and money LOST. The whole field 
of mechanics, every known mechanical movement, and practically every device is covered 
by these two books. If the thing you want has been invented, it is illustrated in them. If 
it hasn’t been invented, then you’ll find in them the nearest things to what you want, some 
movements or devices that will apply in your case, perhaps; or which will give you a key 
from which to work. No book or set of books ever published is of more real value to the 
Inventor, Draftsman, or practical Mechanic than the two volumes described below. 


MECHANICAL MOVEMENTS, POWERS, AND DEVICES. By Gakdnep D. 
Hiscox. 

This is a collection of 1,890 engravings of different mechanical motions and appliances, 
accompanied by appropriate text, making it a book of great value to the inventor, 
the draftsman, and to all readers with mechanical tastes. The book is divided into 
eighteen sections or chapters, in which the subject-matter is classified under the follow¬ 
ing heads: Mechanical Powers; Transmission of Power; Measurement of Power; 
Steam Power; Air Power Appliances; Electric Power and Construction; Navigation 
and Roads; Gearing; Motion and Devic&s; Controlling Motion; Horological; 
Mining; Mill and Factory Appliances; Construction and Devices; Drafting Devices: 
Miscellaneous Devices, etc. 15th edition enlarged. 400 octavo pages. Price . $4.00 


MECHANICAL APPLIANCES, MECHANICAL MOVEMENTS AND NOVEL¬ 
TIES OF CONSTRUCTION. By Gardner D. Hiscox. 

This is a supplementary volume to the one upon mechanical movements. Unlike the 
first volume, which is more elementary in character, this volume contains illustrations 
and descriptions of many combinations of motions and of mechanical dexdces and 
appliances found in different lines of machinery, each device,being shown by a line 
drawing with a description showing its working parts and the method of operation. 
From the multitude of devices described and illustrated might be mentioned, in 
passing, such items as conveyors and elevators, Prony brakes, thermometers, various 
types of boilers, solar engines, oil-fuel burners, condensers, evaporators, Corliss and 
other valve gears, governors, gas engines, water motors of various descriptions, air¬ 
ships, motors and dynamos, automobile and motor bicycles, railway lock signals, 
car couplers, link and gear motions, ball bearings, breech block mechanism for heavy 
gims, and a large accumulation of others of equal importance. 1,000 specially made 
engravings. 396 octavo pages. 4th Edition enlarged. Price.$4.00 

SHOP PRACTICE FOR HOME MECHANICS. By Raymond Francis Yates. 

A thoroughly practical and helpful treatment prepared especially for those who have 
had little or no experience in shop work. The introduction is given over to an ele¬ 
mentary explanation of the fundamentals of mechanical science. This is followed 
by several chapters on the use of small tools and mechanical measuring instruments. 
Elementary and more advanced lathe work is treated in detail and directions given 
for the construction of a number of useful shop appliances. Drilling and reaming, 
heat treatment of tool steel, special lathe operations, pattern maldng, grinding, and 
grinding operations, home foundry work, etc., make up the rest of the volume. The 
book omits nothing that will be of use to those who use tools or to those who wish 
to learn the use of tools The great number of clear engravings (over 300) add 
tremendously to the text matter and to the value of the volume as a visual instructor 
Octavo, 320 pages. 309 engravings. Price , t $3 00 

28 















CATALOGUE OF GOOD, PRACTICAL BOOKS 


MACHINE-SHOP TOOLS AND SHOP PRACTICE. By W. H. Vandebvoobt. 

A work of 552 pages and 672 illustrations, describing in every detail the construction, 
operation, and manipulation of both hand and machine tools. Includes chapters 
on filing, fitting, and scraping surfaces; on drills, reamers, taps, and dies; the lathe 
and its tools; planers, shapers, and their tools; milling machines and cutters; gear 
• cutters and gear cutting; drilling machines and drill work; grinding machines and 
their work; hardening and tempering; gearing, belting, and transmission machinery; 
useful data and tables. 7th Edition. 552 pages. 672 illustrations. Price $4.50 

COMPLETE PRACTICAL MACHINIST. By Joshua Rose. 

The new, twentieth revised and enlarged edition is now ready. This is one of the 
best-known books on macliine-shop work, and written for the practical workman 
, in the language of the workshop. It gives full, practical instructions on the use of 
all kinds of metal-working tools, both hand and machine, and tells how the work 
should be properly done. It covers lathe work, vise work, drills and drilling, taps 
and dies, hardening and tempering, the making and use of tools, tool grinding, mark¬ 
ing out work, machine tools, etc. No machinist’s library is complete without this 
volume. 20th Edition. 547 pages. 432 illustrations. Price .... $3.00 

HENLEY’S ENCYCLOPEDIA OF PRACTICAL ENGINEERING AND ALLIED 
TRADES. Edited by Joseph G. Horner, A.M.I.Mech.E. 

This book covers the entire practice of Civil and Mechanical Engineering. The 
best known experts in all branches of engineering have contributed to these volumes. 
The Cyclopedia is admirably well adapted to the needs of the beginner and the self- 
taught practical man, as well as the mechanical engineer, designer, draftsman, shop 
superintendent, foreman and maclunist. 

It is a modern treatise in five volumes. Handsomely bound in half morocco, each 
volume containing nearly 500 pages, with thousands of illustrations, including dia¬ 
grammatic and sectional drawings with full explanatory details. Five large volumes. 
Price. $30.00 

MODEL MAKING Including Workshop Practice, Design and Construction of 
Models. Edited by Raymond F. Yates. Editor of “Everyday Engineering 
Magazine.” 

This book will help you to become a better mechanic. It is full of suggestions for those 
who like to make things, amateur and professional alike. It has been prepared es¬ 
pecially for men with mechanical hobbies. Some may be engineers, machinists, jew¬ 
elers, pattern makers, office clerks or bank presidents. Men from various walks of 
life have a pecuhar interest in model engineering. Model Making will be a help and 
an inspiration to such men. It tells them “how-to-do” and “how-to-make” things 
in simple, understandable terms. Not only this, it is full of good, clear working 
drawings and photographs of the models and apparatus described. Each model has 
been constructed and actually works if it is made according to directions. 379 pages. 
300 illustrations. Price. $3.00 

ABRASIVES AND ABRASIVE WHEELS. By Fred B. Jacobs. 

A new book for everyone interested in abrasives or grinding. A careful reading of 
the book will not only make mechanics better able to use abrasives intelUgently, but 
it will also tell the shop superintendent of many short cuts and efficiency-increasing 
kinks. The economic advantages in using large grinding wheels are fully explained, 
together with many other things that will tend to give the superintendent or workman 
a keen insight into abrasive engineering. 340 pages. 174 illustrations. This is an 
indispensable book for every machinist. Price.. $3.00 

HOME MECHANIC’S WORKSHOP COMPANION. By Andrew Jackson, Jr. 

This treatise includes a compilation of useful suggestions that cannot fail to interest 
the handy man, and while it is not intended for mechanical experts or scientists, it will 
prove to be a veritable store of information for anyone who desires to rig up a small 
shop where «dd jobs can be carried on. Price. 75 cents 

20 







CATALOGUE OF GOOD, PRACTICAL BOOKS 


MARINE ENGINEERING 


XHE NAVAL ARCHITECT’S AND SHIPBUILDER’S POCKETBOOK. Of 

Formulae, Rules, and Tables and Marine Engineer’s and Surveyor’s Handy 
Book of Reference. By Clement Mackrow and Lloyd Woollard. * 

The twelfth revised and enlarged edition of tliis most comprehensive work has just 
been issued. It is absolutely indispensable to all engaged in the Shipbuilding Industry, 
as it condenses into a compact form all data and formulae^that are ordinarily required. 
The book is completely up to date, including among other subjects a section on 
Aeronautics. 760 pages, limp leather binding. Price.$6.00 

AIARINE ENGINES AND BOILERS—THEIR DESIGN AND CONSTRUC- . 
TION. THE STANDARD BOOK. By Dr. G. Bauer, Leslie S. Robertson 
and S. Bryan Donkin. 

In the words of Dr. Bauer, the present work owes its origin to an oft felt want of a 
condensed treatise embodying the theoretical and practical rules used in designing 
marine engines and boilers. The fact that the original German work was written by 
the chief engineer of the famous Vulcan Works, Stettin, is in itself a guarantee that 
this book is in all respects thoroughly up-to-date, and that it embodies aU the in¬ 
formation which is necessary for the design and construction of the highest types of 
marine engines and boilers. It may be said that the motive power which Dr. Bauer 
has placed in the fast German liners that have been turned out of late years from the 
Stettin Works represent the very best practice in marine engineering of the present 
day. The work is clearly written, thoroughly systematic, theoretically sound; while 
the character of the plans, drawings, tables, and statistics is without reproach. The 
illustrations are careful reproductions from actual working drawings, with some well- 
executed photographic views of completed engines and boilers. Fifth impression. 
744 pages. 550 illustrations, and numerous tables. Cloth. Price . . . $10.00 

MANUAL TRAINING 


ECONOMICS OF MANUAL TRAINING. By Louis Rouillion. 

The only book published that gives just the information needed by all interested iii 
Manual Training, regarding Builaings, Equipment, and Supplies. Shows exactly 
what is needed for all grades of the work from the Kindergarten to the High and 
Normal School. Gives itemized lists of everything used in Manual Training Work 
and tells just what it ought to cost. Also shows where to buy supplies, etc. Contains 
174 pages, and is fully illustrated. 2d edition. Price.$2.00 

MOTOR BOATS 


jMOTOR BOATS AND BOAT MOTORS. By Victor W. Pagie and A. G. Leitch. 

All who are interested in motor boats, either as owners, builders or repairmen, will 
find this latest work a most comprehensive treatise on the design, construction, opera¬ 
tion and repair of motor boats and their power j)lants. It is really two complete 
books in one cover as it consists of two parts, each complete in itself. Part One deals 
with The Hull and Its Fittings, Part Two considers The Power Plant and Its 
Auxiliaries. A valuable feature of this book is the complete set of dimensioned 
working drawings detailing the construction of five different types of boats ranging 
from a 16-foot shallow draft, tunnel stern general utility craft to a 25-foot cabin 
cruiser. These plans are by A. C. Leitch, a practical boat builder and expert naval 
architect, and are complete in every particular. Full instructions are given for the 
selection of a power plant and its installation in the hull. Valuable advice is included 
on boat and engine operation and latest designs of motors are described and illustrated. 
The instructions for overhauling boat and engine are worth many times the small 
cost of the book. It is a comprehensive work of reference for all interested in motor 
boating in any of its phases. Octavo. Cloth. 372 illustrations. 524 pages. 
Price..$4.00. 


30 













CATALOGUE OF GOOD, PRACTICAL BOOKS 


MOTORCYCLES 


MOTORCYCLES AND SIDE CARS, THEIR CONSTRUCTION, MANAGE¬ 
MENT AND REPAIR. By Victor W. Page, M.E. 

The only complete work published for the motorcyclist and repairman. Describes 
fully all leading types of machines, their design, construction, maintenance, operation 
and repair. This treatise outlines fully the operation of two- and four-cycle power 
plants and aU ignition, carburetion and lubrication systems in detail. Describes all 
representative types of free engine clutches, variable speed gears and power trans¬ 
mission systems. Gives complete instructions for operating and repairing all types. 
Considers fully electric self-starting and lighting systems, all types of spring frames 
and spring forks and shows leading control methods. For those desiring technical 
information a complete series of tables and many formulae to assist in designing are 
included. The work tells how to figure power needed to climb grades, overcome air 
resistance and attain high speeds. It shows how to select gear ratios for various 
weights and powers, how to figure braking efficiency required, gives sizes of belts and 
chains to transmit power safely, and shows how to design sprockets, belt pulleys, etc. 
This work also includes complete formulae for figuring horse-power, shows how dyna¬ 
mometer tests are made, defines relative efiQciency of air- and water-cooled engines, plain 
and anti-friction bearings and many other data of a practical, helpful, engineering 
nature. Remember that you get this information in addition to the practical de¬ 
scription and instructions which alone are worth several times the price of the book. 
2nd Edition Revised and Enlarged. 693 pages. 371 specially made illustrations. 
Cloth. Price.$3.00 


WHAT IS SAID OF THIS BOOK: 

" Here is a book that should be in the cycle repairer’s kit .”—American Blacksmith. 

“ The best way for any rider to thoroughly understand his machine, is to get a copy 
of this book; it is worth many times its price .”—Pacific Motorcyclist. 


PATTERN MAKING 


PRACTICAL PATTERN MAKING. By F. W. Barrows. 

This book, now in its second edition, is a comprehensive and entirely practical treatise 
on the subject of pattern making, illustrating pattern work in both wood and metal, 
and with definite instructions on the use of plaster of Paris in the trade. It gives 
specific and detailed descriptions of the materials used by pattern makers and de¬ 
scribes the tools, both those for the bench and the more interesting machine tools; 
having complete chapters on the Lathe, the Circular Saw, and the Band Saw. It gives 
many examples of pattern work, each one fully illustrated and explained with much 
detail. These examples, in their great variety, offer much that will be foimd of 
interest to all pattern makers, and especially to the younger ones, who are seeking 
information on the more advanced branches of their trade. 

In this second edition of the work will be found much that is new, even to those who 
have long practiced this exacting trade. In the description of patterns as adapted 
to the Moulding Machine many difficulties which have long prevented the rapid and 
economical production of castings are overcome; and this great, new branch of the 
trade is given much space. Stripping plate and stool plate work and the less expen¬ 
sive vibrator, or rapping plate work, are all explained in detail. 

Plain, everyday rules for lessening the cost- of patterns, with a complete system of 
cost keeping, a detailed method of marking, applicable to all branches of the trade, 
with complete information showing what the pattern is, its specific title, its cost, 
date of production, material of which it is made, the number of pieces and core¬ 
boxes, and its location in the pattern safe, all condensed into a most complete card 
record, with cross index. 

The book closes with an original and practical method for the inventory and valua¬ 
tion of patterns. 2nd Edition. Containing nearly 350 pages and 170 illustra¬ 
tions. Price.$2.50 


31 










CATALOGUE OF GOOD, PRACTICAL BOOKS 


PERFUMERY 


PERFUMES AND COSMETICS, THEIR PREPARATION AND MANUFAC¬ 
TURE. By G. W. Ajikinson, Perfumer. 


A comprehensive treatise, in which there has been nothing omitted that could be of 
value to the perfumer or manufacturer of toilet preparations. Complete directions 
for making handkerchief perfumes, smelling-salts, sachets, fumigating pastilles; 
preparations for the care of the skin, the mouth, the hair, cosmetics, hair dyes and 
other toilet articles are given, also a detailed description of aromatic substances; their 
nature, tests of purity, and wholesale manufacture, including a chapter on synthetic 
products, with formulas for their use. A book of general, as well as professional in¬ 
terest, meeting the wants not only of the druggist and perfume manufacturer, but 
also of the general public. Fourth Edition much enlarged and brought up-to-date. 
Nearly 400 pages, illustrated. Price. $6.00 


WHAT IS SAID OP THIS BOOK: 


*• The most satisfactory work on the subject of Perfumery that we have ever seen. 

“ We feel safe in saying that here is a book on Perfumery that will not disappoint you, 
for it has practical and excellent formulae that are within your ability to prepare 
readily. 

“ We recommend the volume as worthy of confidence, and say that no purchaser will be 
disappointed in securing from its pages good value for its cost, and a large dividend 
on the same, even if he should use but one per cent of its working formidae. There 
is money in it for every user of its information.”— Pharmaceutical Record. 


PLUMBING 


MECHANICAL DRAWING FOR PLUMBERS. By R. M. Starbuck. 


A concise, comprehensive and practical treatise on the subject of mechanical drawing 
in its various modern applications to the work of all who are in any way connected 
with the plumbing trade. Nothing will so help the plumber in estimating and in 
explaining work to customers and workmen as a knowledget.of drawing, and to the 
workman it is of inestimable value if he is to rise above his position to positions of 
greater responsibility. Among the chapters contained are: 1. Value to plumber of 
knowledge of drawing; tools required and their use; common views needed in mechan¬ 
ical drawing. 2. Perspective versus mechanical drawing in showing plumbing con¬ 
struction. 3. Correct and incorrect methods in plumbing drawing; plan and elevation 
explained. 4. Floor and cellar plans and elevation; scab drawings; use of triangles 
5. Use of triangles; drawing of fittings, traps, etc. 6. Drawing plumbing elevations 
and fittings. 7. Instructions in drawing plumbing elevations. 8. The drawing of 
plumbing fixtures; scale drawings. 9. Drawings of fixtures and fittings. 10 Inking 
of drawings. 11. Shading of drawings. 12. Shading of drawings. 13 SectioniU 
drawings; drawing of threads. 14. Piumbing elevations from architect’s r>isir. i c; 


CilCO ... 



$ 2.00 


32 










CATALOGUE OF GOOD, PRACTICAL BOOKS 


MODERN PLUMBING ILLUSTRATED. By R. M. Starbuce. 

This book represents the highest standard of plumbing work. It has been adopted 
and used as a reference book by the United States Government, in its sanitary work in 
Cuba, Porto Rico, and the Phihppines, and by the principal Boards of Health of the 
United States and Canada. 

It gives connections, sizes and working data for all flxtur^ and groups of fixtures. It 
is helpful to the master plumber in demonstrating to his customers and in figuring 
work. It gives the mechanic and student quick and easy access to the best modern 
plumbing practice. Suggestions for estimating plumbing construction are contained 
in its pages. This book represents, in a word, the latest and best up-to-date practice 
and should be in the hands of every architect, sanitary engineer and plumber who 
wishes to keep himself up to the minute on this important feature of construction. 
Contains following chapters, each illustrated with a full-page plate: Kitchen sink, 
laundry tubs, vegetable wash sink; lavatories, pantry sinks, contents of marble slabs; 
bath tub, foot and sitz bath, shower bath; water closets, venting of water closets; low* 
down water closets, water closets operated by flush valves, water closet range; slop sink, 
urinals, the bidet; hotel and restaurant sink, grease trap; refrigerators, safe wastes, laun¬ 
dry waste, lines of refrigerators, bar sinks, soda fountain sinks; horse stall, frost-proof 
water closets; connections for S traps, venting; connections for drum traps; soil pipe 
connections; supporting of soil pipe; main trap and fresh air inlet; floor drains and 
cellar drains, subsoil drainage; water closets and floor connections; local venting; 
connections for bath rooms; connections for bath rooms, continued; connections for 
bath rooms, continued; connections for bath rooms, continued; examples of poor 
practice; roughing work ready for test; testing of plumbing system; method of con¬ 
tinuous venting; continuous venting for two-floor work; continuous venting for two 
lines of fixtures on three or more floors; continuous venting of water closets; plumb¬ 
ing for cottage house; construction for cellar piping; plumbing for residence, use of 
special fittings; plumbing for two-flat house; plumbing for apartment building, plumb¬ 
ing for double apartment building; plumbing for office building; plumbing for public 
toilet rooms; plumbing for pubUc toilet rooms, continued; plumbing for bath estab¬ 
lishment; plumbing for engine house, factory plumbing; automatic flushing for 
schools, factories, etc.; use of flushing valves; urinals for public toilet rooms; the 
Durham system, the destruction of pipes by electrolysis; construction of work without 
use of lead; automatic sewage lift; automatic sump tank; country plumbing; construc¬ 
tion of cesspools; septic tank and automatic sewage siphon; country plumbing; water 
supply for country house; thawing of water mains and service by electricity; double 
boilers; hot water supply of large buildings; automatic control of hot water tank; sug¬ 
gestion for estimating plumbing construction. 407 octavo pages, fully illustrated by 58 
full-page engravings. Third, revised and enlarged edition just issued. Price . $5.00 

STANDARD PRACTICAL PLUMBING. By R. M. Starbuck. 

A complete practical treatise of 450 pages covering the subject of Modern Plumbing 
in all its branches, a large amount of space being devoted to a very complete and 
practical treatment of the subject of Hot Water Supply and Circulation and Range 
Boiler Work. Its thirty chapters include about every phase of the subject one can 
think of, making it an indispensable work to the master plumber, the journeyman 
plumber, and the apprentice plumber, containing chapters on: the plumber’s tools; 
wiping solder; composition and use; joint wiping: lead work; traps; siphonage of 
traps; venting; continuous venting; house sewer and sewer connections: house drain; 
soil piping, roughing; main trap and fresh air inlet; floor, yard, cellar drains, rain 
leaders, etc.; fixture wastes; water closets; ventilation; improved plumbing connec¬ 
tions; residence plumbing; plumbing for hotels, schools, factories, stables, etc.; 
modern country plumbing; filtration of sewage and water supply; hot and cold 
supply: range boilers; circulation: circulating pipes; range boiler problems; hot 
water for large buildings: water lift and its use; multiple connections for hot water 
boilers; heating of radiation by supply system; theory for the plumber; drawing for 
the plumber. Fully illustrated by 347 engravings. Price. $3.50 


33 






CATALOGUE OF GOOD, PRACTICAL BOOKS 


RECIPE BOOK 


HENLEY’S TWENTIETH CENTURY BOOK OF RECIPES, FORMULAS AND 
PROCESSES. Edited by Gardner D. Hiscox. 

The most valuable Techno-chemical Formula Book published, including over 10,000 
selected scientific, chenfical, technological, and practical recipes and processes. 

This is the most complete Book of Formulas ever published, giving thousands of 
recipes for the manufacture of valuable articles for everyday use. Hints, Helps, 
Practical Ideas, and Secret Processes are revealed within its pages. It covers every 
branch of the useful arts and tells thousands of ways of making money, and is just the 
book everyone should have at his command. 

Modern in its treatment of every subject that properly falls within its scope, the book 
may truthfully be said to present the very latest formulas to be found in the arts and 
industries, and to retain those processes which long experience has proven worthy of a 
permanent record. To present here even a limited number of the subjects which find 
a place in this valuable work would be difficult. Suffice to say that in its pages will 
be found matter of intense interest and immeasurably practical value to the scientific 
amateur and to him who wishes to obtain a knowledge of the many processes used in 
the arts, trades and manufacture, a knowledge which will render his pursuits more 
instructive and remunerative. Serving as a reference book to the small and large 
manufacturer and supplying intelligent seekers with the information necessary to 
conduct a process, the work wifi be found of inestimable worth to the Metallurgist, the 
Photographer, the Perfumer, the Painter, the Manufacturer of Glues, Pastes, Cements, 
and Mucilages, the Compounder of Alloys, the Cook, the Physician, the Druggist, the 
Electrician, the Brewer, the Engineer, the Foundryman, the Machinist, the Potter, the 
Tanner, the Confectioner, the Chiropodist, the Manicure, the Manufacturer of Chem¬ 
ical Novelties and Toilet Preparations, the Dyer, the Electroplater, the Enameler, the 
Engraver, the Provisioner, the Glass Worker, the Goldbeater, the Watchmaker, the 
Jeweler, the Hat Maker, the Ink Manufacturer, the Optician, the Farmer, the Dairy¬ 
man, the Paper Maker, the Wood and Metal Worker, the Chandler and Soap Maker, 
the Veterinary Surgeon, and the Technologist in general. 

A mine of information, and up-to-date in every respect. A book which will prove of 
value to EVERYONE, as it covers every branch of the Useful Arts. Every home 
needs this book; every office, every factory, every store, every public and private en¬ 
terprise— EVERYWHERE — should have a copy. 800 pages. Cloth Bound. 
Price. $4.00 


> WHAT IS SAID OF THIS BOOK: 

*‘Your Twentieth Century Book of Recipes, Formulas, and Processes duly received. 
I am glad to have a copy of it, and if I could not replace it, money couldn’t buy it. It 
is the best thing of the sort I ever saw.” (Signed) M. E. Trux, Sparta, Wis. 

” There are few persons who would not be able to find in the book some single formula 
that would repay several times the cost of the book.”— Merchants' Record and Show 
Window. 

“ I purchased your book ‘ Henley’s Twentieth Century Book of Recipes, Formulas and 
Processes’ about a year ago and it is worth its weight in gold.” —Wm. H. Murray, 
Bennington, Vt. 

•THE BOOK WORTH THREE HUNDRED DOLLARS” 

“On close examination of your ‘Twentieth Century Receipt Book,’ I find it to be a 
very valuable and useful book with the very best of practical information obtainable. 
The price of the book, $4.00, is very small in comparison to the benefits which one can 
obtain from it. I consider the book worth fully three hundred dollars to anyone.” 
—Dr. a. C. Spetts, New York. 

‘‘ONE OF THE WORLD’S MOST USEFUL BOOKS” 

"Some time ago, I got one of your ‘Twentieth Century Books of Formulas’' and have 
made my livung from it ever since. I am alone since my husband’s death with two 
small children to care for and am trying so hard to support them. I have customers 
who take from me Toilet Articles I put up, following directions given in the book, 
and I have found every one of them to be fine.’’-—M rs. J. H. McMaken, West Toledo, 


34 







CATALOGUE OF GOOD, PRACTICAL BOOKS 


RUBBER 


RUBBER HAND STAMPS AND THE MANIPULATION OF INDIA RUBBER. 

By T. O’CoNOK Sloane. 

This book gives full details on all points, treating in a concise and simple manner the 
elements of nearly everything it is necessary to understand for a commencement in 
any branch of the India Rubber Manufacture. The making of aU kinds of Riibber 
Hand Stamps, Small Articles of India Rubber, U. S. Government Composition, Dating 
Hand Stamps, the Manipulation of Sheet Rubber, Toy Balloons, India Rubber Solu¬ 
tions, Cements, Blackings, Renovating Varnish, and Treatment for India Rubber 
Shoes, etc.; the Hektograph Stamp Inks, and Miscellaneous Notes, with a Short 
Account of the Discovery, Collection and Manufacture of India Rubber, are set forth 
in a manner designed to be readily understood, the explanations being plain and simple. 
Including a chapter on Rubber Tire Making and Vulcanizing; also a chapter on the 
uses of rubber in Surgery and Dentistry. Third revised and enlarged edition. 175 
pages. Illustrated. $1.50 


SAWS 


SAW FILING AND MANAGEMENT OF SAWS. By Robert Grimshaw. 

A practical hand-book on filing, gumming, swaging, hammering, and the brazing of 
band saws, the speed, work, and power to run circular saws, etc. A handy book for 
those who have charge of saws, or for those mechanics who do their own filing, as it deals 
with the proper shape and pitches of saw teeth of all kinds and gives many useful hints 
and rules for gumming, setting, and filing, and is a practical aid to those who use saws 
for any purpose. Complete tables of proper shape, pitch, and saw teeth as well as 
sizes and number of teeth of various saws are included. Fourth edition, revised and 
enlarged. Illustrated. Price. .... $1.50 


SCREW CUTTING 


THREADS AND THREAD-CUTTING. By Colvin and Stable. 

This clears up many of the mysteries of tliread-cutting, such as double and triple 
threads, internal threads, catching threads, use of hobs, etc. Contains a lot of useful 
hints and several tables. Fourth Edition, Price 85 cents 

STEAM ENGINEERING 


MODERN STEAM ENGINEERING IN THEORY AND PRACTICE. By 

Gardner D. Hiscox. 

This is a complete and practical work issued for Stationary Engineers and Firemen, 
dealing with the care and management of boilers, engines, pumps, superheated steam, 
refrigerating machinery, dynamos, motors, elevators, air compressors, and ail other 
branches with which the modern engineer must be familiar. Nearly 200 questions with 
their answers on steam and electrical engineering, likely to be asked by the Examin¬ 
ing Board, are included. 

Among the chapters are: Historical: steam and its properties; appliances for the 
generation of steam; types of boilers; chimney and its work; heat economy of the 
feed water; steam pumps and their work; incrustation and its work; steam above 
atmospheric pressure; flow of steam from nozzles; superheated steam and its work; 
adiabatic expansion of steam; indicator and its work; steam engine proportions: slide 
valve engines and valve motion; Corliss engine and its valve gear; compound engine 
and its theory; triple and multiple expansion engine: steam turbine; refrigeration: 
elevators and their management: cost of power; steam engine troubles; electric 
power and electric plants. 487 pages. 405 engravings. 3d Edition. . . . $3.50 

35 


















CATALOGUE OF GOOD, PRACTICAL BOOKS 


AMERICAN STATIONARY ENGINEERING. By W. E. Crane. 

This book begins at the boiler room and takes in the whole power plant. A plain 
talk on every-day work about engines, boilers, and their accessories. It is not intended 
to be scientific or mathematical. All formulas are in simple form so that anyone 
understanding plain arithmetic can readily understand any of them. The author 
has made this the most practical book in print; has given the results of his years of 
experience, and has included about all that has to do with an engine room or a power 
plant. You are not left to guess at a single point. You are shown clearly what to 
expect under the various conditions; how to secure the best results; ways of prevent¬ 
ing “shut downs” and repairs; in short, all that goes to make up the requirements 
of a good engineer, capable of taking charge of a plant. It’s plain enough for practical 
men and yet of value to those high in the profession. 

A partial list of contents is; The boiler room, cleaning boilers, firing, feeding; pumps, 
inspection and repair; chimneys, sizes and cost; piping; mason work; foimdations; 
testing cement; pile driving; engines, slow and high speed; valves; valve setting; 
CorUss engines, setting valves, single and double eccentric; air pumps and condensers; 
different types of condensers; water needed; lining up; pounds; pins not square in 
crosshead or crank; engineers’ tools; pistons and piston rings; bearing metal; hard¬ 
ened copper; drip pipes from cyhnder jackets; belts, how made, care of; oils; greases; 
testing lubricants; rules and tables, including steam tables; areas of segments; 
squares and square roots; cubes and cube root; areas and circumferences of circles. 
Notes on: Brick work; explosions; pumps; pump valves; heaters, economizers; 
safety valves; lap, lead, and clearance. Has a complete examination for a license, 
etc., etc. Third edition. 311 pages. 131 Illustrations. Price. . . . $2.50 


ENGINE RUNNER’S CATECHISM. By Robert Grimshaw. 

A practical treatise for the stationary engineer, telling how to erect, adjust, and run 
the principal steam engines in use in the United States. Describing the principal 
features of various special and well-known makes of engines: Temper Cut-off, Shipping 
and Receiving Foundations, Erecting and Starting, Valve Setting, Care and Use, 
Emergencies, Erecting and Adjusting Special Engines. 

The questions asked throughout the catechism are plain and to the point, and the 
answers are given in such simple language as to be readily understood by anyone. All 
the instructions given are complete and up-to-date; and they are written in a popular 
style, without any technicalities or mathematical formulae. The work is of a handy 
size for the pocket, clearly and well printed, nicely bound, and profusely illustrated. 

To young engineers this catechism will be of great value, especially to those who may 
be preparing to go forward to be examined for certificates of competency; and to 
engineers generally it will be of no little service, as they will find in this volume more 
really practical and useful information than is to be found anywhere else within a like 
compass. 387 pages. Seventh edition. Price. $2.00 


HORSE-POWER CHART. 

Shows the horse-power of any stationary engine without calculation. No matter what 
the cyhnder diameter of stroke, the steam pressure of cut-off, the revolutions, or 
whether condensing or non-condensing, it’s all there. Easy to use, accurate, and 
saves time and calculations. Especially useful to engineers and designers. 50 cents 

STEAM ENGINE CATECHISM. By Robert Grimshaw. 

Tins unique volume of 413 pages is not only a catechism on the question and answer 
principle, but it contains formulas and worked-out answers for all the Steam problems 
that appertain to the operation and management of the Steam Engine. Illustrations 
of various valves and valve gear with their principles of operation are given. Thirty- 
four Tables that are indispensable to every engineer and fireman that wishes to be 
progressive and is ambitious to become master of his calling are within its pages. It is 
a most valuable instructor in the service of Steam Engineering. Leading engineers 
have recommended it as a valuable educator for the beginner as well as a reference book 
for the engineer. It is thoroughly indexed for every detail. Every essential question 
on the St^m Engine with its answer is contained in this valuable work. Sixteenth 
edition. Price...$2.00 






CATALOGUE OF GOOD, PRACTICAL BOOKS 


STEAM ENGINEER’S ARITHMETIC. By Colvin-Chenet. 

A practical pocket-book for the steam engineer. Shows how to work the problems of 
the engine room and shows “why.” Tells how to figure horse-power of engines and 
boilers; area of boilers; has tables of areas and circumferences; steam tables; has a 
dictionary of engineering terms. Puts you on to all of the little kinks in figuring what¬ 
ever there is to figure around a power plant. Tells you about the heat unit; absolute 
zero; adiabatic expansion; duty of engines; factor of safety; and a thousand and one 
other things; and everything is plain and simple—not the hardest way to figure, but 
the easiest. Second Edition. 75 cents 

STEAM ENGINE TROUBLES. By H. Hamkens. 

It is safe to say that no book has ever been published which gives the practical en¬ 
gineer such valuable and comprehensive information on steam engine design and 
troubles. 

Not only does it describe the troubles the principal parts of steam engines are subject 
to; it contrasts good design with bad, points out the most suitable material for certain 
parts, and the most approved construction of the same; it gives directions for correct¬ 
ing existing evils by following which breakdowns and costly accidents can be avoided. 
Just look into 1/he nature of the information tliis book gives on the following sub¬ 
jects. There are descriptions of cyUnders, valves, pistons, frames, pillow blocks and 
other bearings, connecting rods, wristplates, dashpots, reachrods, valve gears, gover¬ 
nors, piping, throttle and emergency valves, safety stops, fiy-wheels, oilers, etc. If 
there is any trouble with these parts, the book gives you the reasons and tells how to 
remedy them. 

The principal considerations in the building of foundations are given with the size, 
area and weight required for the same, also the setting of templets and lining up, and 
a complete account of the erection and "breaking in” of new engines in the language 
of the man on the job. 

Contains special chapters on: I. Cylinders. II. Valves. III. Piping and Separa¬ 
tors. IV. Throttle and Emergency Valves. V. Pistons. VI. Frames. VII. Bear¬ 
ings. VIII. Connecting Rods. IX. Hookrods. X. Dashpots. XI. Governors. 
XII. Releasing Gears. XIII. Wristplates and Valve Motions. XIV. Rodends and 
Bonnets. XV. Oilers. XVI. Receivers. XVII. Foundations. XVIIl. Erection. 
XIX. Valve-Setting. XX. Operation. 284 pages. 276 illustrations. Price $2.50 


STEAM HEATING AND VENTILATION 


PRACTICAL STEAM, HOT-WATER HEATING AND VENTILATION. By 

A. G. King. 

This book is the standard and latest work published on the subject and has been pre¬ 
pared for the use of all engaged in the business of steam, hot-water heating, and ventila¬ 
tion. It is an original and exhaustive work. Tells how to get heating contracts, how 
to install heating and ventilating apparatus, the best business methods to be used, 
with “Tricks of the Trade” for shop use. Rules and data for estimating radiation 
and cost and such tables and information as make it an indispensable work for every¬ 
one interested in steam, hot-water heating, and ventilation. It describes all the principal 
systems of steam, hot-water, vacuum, vapor, and vacuum-vapor heating, together 
with the new accelerated systems of hot-water circulation, including chapters on 
up-to-date methods of ventilation and the fan or blower system of heating and ventila¬ 
tion. Containing chapters on: I. Introduction. II. Heat. III. Evolution of 
artificial heating apparatus. IV. Boiler surface and settings. V. The chimney fine. 
VI. Pipe and fittings. VII. Valves, various kinds. VIII. Forms of radiating 
surfaces. IX. Locating of radiating surfaces. X. Estimating radiation. XI. Steam¬ 
heating apparatus. XII. Exhaust-steam heating. XIII. Hot-water heating. XIV. 
Pressure systems of hot-water work. XV. Hot-water appliances. XVI. Greenhouse 
heating. XVII. Vacuum vapor and vacuum exhaust heating. XVIII. Miscella¬ 
neous heating. XIX. Radiator and pipe connections. XX. Ventilation. XXI. 
Mechanical ventilation and hot-blast heating. XXII. Steam appUances. XXIII. 
District heating. XXIV. Pipe and boiler covering. XXV. Temperature regulation 
and heat control. XXVI. Business methods. XXVII. Miscellaneous. XXVIII. 
Rules, tables, and useful information. 402 pages. 300 detailed engravings. Third 
Edition—Revised. Price.$3.5^ 


37 









CATALOGUE OF GOOD, PRACTICAL BOOKS 


500 PLAIN ANSWERS TO DIRECT QUESTIONS ON STEAM, HOT-WATER, 
VAPOR AND VACUUM HEATING PRACTICE. By Alfred G. King, 

This work, just off the press, is arranged in question and answer formit is intended as 
a guide and text-book for the younger, inexperienced fitter and as a reference book for 
all fitters. This book tells “how” and also tells “why.” No work of its kind has 
ever been published. It answers all the questions regarding each method or system 
that would be asked by the steam fitter or heating contractor, and may be used as a 
text or reference book, and for examination questions by Trade Schools or Steam 
Fitters’ Associations. Rules, data, tables and descriptive methods are given, to¬ 
gether with much other detailed information of daily practical use to those engaged in 
or interested in the various methods of heating. Valuable to those preparing for 
examinations. Answers every question asked relating to modern Steam, Hot-Water, 
Vapor and Vacuum Heating. Among the contents are: The Theory and Laws of 
Heat. Methods of Heating. Chimneys and Flues. Boilers for Heating. Boiler 
Trimmings and Settings. Radiation. Steam Heating. Boiler, Radiator and Pipe 
Connections for Steam Heating. Hot Water Heating. The Two-Pipe Gravity 
System of Hot Water Heating. The Circuit System of Hot Water Heating. The 
Overhead System of Hot Water Heating. Boiler, Radiator and Pipe Connections for 
Gravity Systems of Hot Water Heating. Accelerated Hot Water Heating. Ex¬ 
pansion Tank Connections. Domestic Hot Water Heating. Valves and Air Valves. 
Vacuum Vapor and Vacqo-Vapor Heating. Mechanical Systems of Vacuum Heating. 
Non-Mechanical Vacuum Systems. Vapor Systems. Atmospheric and Modulating 
Systems. Heating Greenhouses. Information, Rules and Tables. 214 pages, 127 
illustrations. Octavo, Cloth. Price.$2.50 

STEEL 


STEEL: ITS SELECTION, ANNEALING, HARDENING, AND TEMPERING. 

By E. R. Markham. 

This book tells how to select, and how to work, temper, harden, and anneal steel for 
everything on earth. It doesn’t tell how to temper one class of tools and then leave 
the treatment of another kind of tool to your imagination and judgment, but it gives 
careful instructions for every detail of every tool, whether it be a tap, a reamer or just 
a screw-driver. It tells about the tempering of small watch springs, the hardening of 
cutlery, and the annealing of dies. In fact, there isn’t a thing that a steel worker 
would want to know that isn’t included. It is the standard book on selecting, harden¬ 
ing, and tempering all grades of steel. Among the chapter headings might be mentioned 
the following subjects: Introduction: the workman; steel; methods of heating; 
heating tool steel; forging; annealing; hardening baths; baths for hardening; harden¬ 
ing steel; drawing the temper after hardening; examples of hardening; pack harden¬ 
ing; case hardening: spring tempering; making tools of machine steel; special steels; 
steel for various tools, causes of trouble; high speed steels, etc. 400 pages. Very 
fully illustrated. Foiu-th Edition. Price.’ . $3.00 

HARDENING, TEMPERING, ANNEALING, AND FORGING OF STEEL. 
INCLUDING HEAT TREATMENT OF MODERN ALLOY STEELS. By 

J. V. Woodworth. 

A new work treating in a clear, concise manner all modern processes for the heating, 
annealing, forging, welding, hardening, and tempering of high and low grade steel, 
making it a book of great practical value to the metal-working mechanic in general, 
with special directions for the successful hardening and tempering of all steel tools 
used in the arts, including milling cutters, taps, thread dies, reamers, both solid and 
shell, hollow milks, punches and dies, and all kinds of sheet metal working tools, shear 
blades, saws, fine cutlery, and metal cutting tools of all description, as well as for all 
implements of steel both large and small. In this work the simplest and most satis¬ 
factory hardening and tempering processes are given. 

The uses to which the leading brands of steel may be adapted are concisely presented, 
and their treatment for working under different conditions explained, also the special 
methods for the hardening and tempering of special brands. 

A chapter devoted to the different processes for case-hardening is also included, and 
special reference made to the adaptation of machinerj' steel for tools of various kinds. 
5th Edition. 321 pages. 20i illustrations, -^rice.. . $3.00 










CATALOGUE OF GOOD, PRACTICAL BOOKS 


TRACTORS 


MODERN GAS TRACTOR: ITS CONSTRUCTION, UTILITY, OPERATION 
AND REPAIR. By Victor W. Page. 

An enlarged and revised edition that treats exclusively on the design and construction 
of farm tractors and tractor power plants, and gives complete instructions on their 
care, operation and repair. All types and sizes of gasoline, kerosene and oil tractors 
are described, and every phase of traction engineering practice fully covered. In¬ 
valuable to all desiring reliable information on gas motor propelled traction engines 
and their use. All new 1921 types of tractors are described and complete instructions 
are given for their use on tlie farm. The chapter on engine repairir^ has been greatly 
enlarged and complete and detailed instructions are now given for repairing weli 
known and videly u.sed tractor pov er plants, numeroiis new forms of which are 
described. Valuable information compiled by Government experts on laying out 
fields for tractor plowing and numerous practical suggestions for hitches so all types 
of agricultural machinery can be operated by tractors, are outlined. The chapter 
on tractor construction and upkeep has been more than doubled in size. Over 100 
new illustrations have been added and the book greatly enlarged. Full instructions 
are now given for using kerosene and distillate as fuel. The 1921 edition is 50 per 
cent larger than the second edition and is more than ever the acknowledged authority 
on farm tractors and their many uses. 5 M x 7 H inches. Cloth, nearly 700 pages, 
and about 300 illustrations, 3 folding plates. Price. $3.00 


WELDING 


MODERN WELDING METHODS. By Victor W. Page. 

One of the most instructive books on all methods of joining metals yet published for 
the mechanic and practical man. It considers in detail oxy-acetylene welding, the 
Thermit process and all classes of electric arc and resistance welding. It shows all. 
the apparatus needed and how to use it. It considers the production of welding gases, 
construction and operation of welding and cutting torches of all kmds. It details 
the latest approved methods of preparing work for welding. All forms of gas and 
electric welding machines are described and complete instructions are given for 
installing electric spot and butt welders. Cost data are given and all methods of 
doing the work economically are described. It includes instructions for forge and 
dip brazing and manufacture of hard solders and spelters. It shows and explains 
soft soldering processes and tells how to make solders for any use. Complete instruc¬ 
tions are given for soldering aluminum and authoritative formulas for aluminum solders 
are included. 292 pages. 200 illustrations. 1921 edition. Price . . $3.00 

AUTOMOBILE WELDING WITH THE OXY-ACETYLENE FLAME. By 

M. Keith Dunham. 

Explains in a simple manner apparatus to be used, its care, and how to construct 
necessary shop equipment. Proceeds then to the actual welding of all automobile 
parts, in a manner understandable by everyone. Gives principles never to be for¬ 
gotten. This book is of utmost value, since the perplexing problems arising when 
metal is heated to a melting point are fully explained and the proper methods to 
overcome them shown. 167 pages. Fully illustrated. Price.$1.50 


39 









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